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Jupiter aurLlris Atoimx 

TELESCOPIC Yimr OF THE FILL MO OX 



AN 



INTRODUCTION 



TO 



ASTRONOMY; 



DESIGNED AS A 



TEXT BOOK 



FOR THE 



STUDENTS OF YALE COLLEGE 



SECOND EDITION. 



BY DENISON OLMSTED, A. M., 

PROFESSOR OF NATURAL PHILOSOPHY AND ASTRONOMY. 



NEW YORK: 

COLLINS, KEESE & CO. 

1841. 






Entered, according to Act of Congress, in the year 1839, 

By Denison Olmsted, 
In the Clerk's Office of the District Court of Connecticut. 



Stereotyped by 

RICHARD C. VALENTINE, 

45 Gold-street, New York. 



Printed by B. L. Hamlen, New Haven, Conn. 




PREFACE 



Nearly all who have written Treatises on Astronomy, designed for young 
learners, appear to have erred in one of two ways ; they have either disre- 
garded demonstrative evidence, and relied on mere popular illustration, or they 
have exhibited the elements of the science in naked mathematical formulae. 
The former are usually diffuse and superficial ; the latter, technical and ab- 
struse. 

In the following Treatise, we have endeavored to unite the advantages of 
both methods. We have sought, first, to establish the great principles of 
astronomy on a mathematical basis ; and, secondly, to render the study inter- 
esting and intelligible to the learner, by easy and familiar illustrations. We 
would not encourage any one to believe that he can enjoy a full view of the 
grand edifice of astronomy, while its noble foundations are hidden from his 
sight; nor would we assure him that he can contemplate the structure in its 
true magnificence, while its basement alone is within his field of vision. We 
would, therefore, that the student of astronomy should confine his attention 
neither to the exterior of the building, nor to the mere analytic investigation 
of its structure. We would desire that he should not only study it in models 
and diagrams, and mathematical formulas, but should at the same time acquire 
a love of nature herself, and cultivate the habit of raising his views to the 
grand originals. Nor is the effort to form a clear conception of the motions and 
dimensions of the heavenly bodies, less favorable to the improvement of the 
intellectual powers, than the study of pure geometry. 

But it is evidently possible to follow out all the intricacies of an analytical 
process, and to arrive at a full conviction of the great truths of astronomy, and 
yet know very little of nature. According to our experience, however, but few 
students in the course of a liberal education will feel satisfied with this. They 
do not need so much to be convinced that the assertions of astronomers are 
true, as they desire to know what the truths are, and how they were ascer- 
tained ; and they will derive from the study of astronomy little of that moral 
and intellectual elevation which they had anticipated, unless they learn to look 
upon the heavens with new views, and a clear comprehension of their won- 
derful mechanism. 

Much of the 'difficulty that usually attends the early progress of the astro- 
nomical student, arises from his being too soon introduced to the most perplex- 
ing part of the whole subject, — the planetary motions. In this work, the con- 
sideration of these is for the most part postponed until the learner has become 
familiar with the artificial circles of the sphere, and conversant with the celes- 
tial bodies. We then first take the most simple view possible of the planetary 
motions by contemplating them as they really are in nature, and afterwards 
proceed to the more difficult inquiry, why they appear as they do. Probably 
no science derives such signal advantage from a happy arrangement, as as- 
tronomy ; — an order, which brings out every fact or doctrine of the science just 
in the place where the mind of the learner is prepared to receive it. 

Although we have found it convenient to defer the consideration of the fixed 
stars to a late period, yet we would earnestly recommend to the student to be- 
gin to learn the constellations, and the stars of the first magnitude at least, as 



IV PREFACE. 

soon as he enters upon the study of astronomy. A few evenings spent in this 
way, assisted, where it is practicable, by a friend already conversant with the 
stars, will inspire a higher degree of enthusiasm for the science, and render its 
explanations more easily understood. 

It is recommended to the learner to make a free use of the Analysis, espe- 
cially in reviewing the ground already traversed. If by repeated recurrence to 
these heads, he associates with each a train of ideas, carrying along with him, 
as he advances, all the particulars indicated in these hints, he will secure to 
them an indelible place in his memory. 

With such aids at hand, as Newton, La Place, and Delambre, to expound 
the laws of astronomy, and such popular writers as Ferguson, Biot, and Fran- 
coeur, to supply familiar illustrations of those laws, it might seem an easy task 
to prepare a work like the present ; but a text book made up of extracts from 
these authors, would be ill suited to the wants of our students. We have 
deemed it better therefore, first, to acquire the clearest views we were able of 
the truths to be unfolded, both from an extensive perusal of standard authors, 
and from diligent reflection, and then to endeavor to transfuse our own im- 
pressions into the mind of the learner. Writers of profound attainments in 
astronomy, and of the highest reputation, have often failed in the preparation 
of elementary works, because they lacked one qualification — the experience of 
the teacher. Familiar as they were with the truths of the science, but unac- 
customed to hold communion with young pupils, they were incapable of ap- 
prehending the difficulty and the slowness with which these truths make their 
entrance into the mind for the first time. Even when they attempt to feel 
their way into young minds, by assuming the garb of the instructor, and em- 
ploying popular illustrations, they often betray their want of the experience 
and art of the professional teacher. 

Astronomy, in its grandest and noblest conceptions, addresses itself alike to 
the intellect and to the heart. It demands the highest efforts of the one and the 
warmest and most devout affections of the other, in order fully to comprehend 
its truths and to relish its sublimity. The task of learning the bare elements 
of this, as well as of every other science, is purely intellectual, and is to be re- 
garded only as preparing the way for that more enlarged and exalted contem- 
plation of the heavenly bodies, to which the mind will naturally rise, when it 
can view all things in their true relations to each other. It is therefore essen- 
tial to this study, as a part of a public education, that the student, after ac- 
quiring a knowledge of the elements of the science, should return to the sub- 
ject, and trace the great discoveries of astronomy, as they have succeeded one 
another from the earliest ages of society down to the present time, viewing them 
in connection with the many interesting historical and biographical incidents 
which attended their development. The author is therefore accustomed, in 
his own course of instruction, to follow the study of this "Introduction," with 
a course of Lectures adapted to such a purpose ; and, with similar views, he 
has prepared a volume of "Letters on Astronomy," where he has attempted 
to connect with the leading truths of the science such historical incidents and 
moral reflections, as may at once interest the understanding and amend the 
heart 



ANALYSIS 



DESIGNED AS A BASIS FOR REVIEW AND EXAMINATION. 



PRELIMINARY OBSERVATIONS. 

Page. 

Astronomy defined, 1 

Descriptive Astronomy, 1 

Physical do. 1 

Practical do. 1 

History. — Ancient nations who cultiva- 
ted astronomy, 1 

Pythagoras — his age and country, 1 

His views of the celestial motions, 1 

Alexandrian School — when founded — 
by whom — introduction of astronomi- 
cal instruments, 2 

Hipparchus — his character, 2 

Ptolemy — the Almagest, 2 

Copernicus, Tycho Brahe, Kepler and 

Galileo — respective labors of each,.... 2 

Sir Isaac Newton — his great discovery, 2 

La Place — Mecanique Celeste, 2 

Astrology — Natural and Judicial — ob- 
ject of each, 2 

Accuracy aimed at by astronomers, 3 

Copemican System — its leading doc- 
trines, 3 

Plan of this work, 3 

Part I.— OF THE EARTH. 
Chapter 1. — of the figure and dimensions 

OF THE EARTH, AND THE DOCTRINE OF THE 
SPHERE. 

Figure of the earth, ' 

Proofs, 4 

Dip of the horizon, 4 

How found, 5 

Table of the dip — its use, C 

Exact figure of I he earth, 6 

Its circumference, 6 

Small inequalities of the earth's surface, 6 

Diameter of the earth how determined, 7 
How to divest the mind of preconceived 

erroneous notions, 8 

Doctrine ok the Sphere, defined, 9 

Great and small circles defined, 9 

Axis of a circle — pole, 9 

Situation of the poles of two great cir- 
cles which cut each other at right an- 



g'os, 



Punts of intersection of two great cir- 
cles — how many degrees apart, 10 



Page. 
When a great circle passes through the 

pole of another, how does it cut it ? . 10 

Secondary defined, 10 

Angle made by two great circles how 

measured, 10 

Terrestrial and Celestial spheres distin- 
guished, 10 

Horizon defined, 1 

Sensible horizon, 1 

Rational do 1 

Zenith and Nadir, 1 

Vertical circles, 1 

Meridian 1 

Prime Vertical, 1 

How the place of a celestial body is de- 
termined, 11 

Altitude — azimuth — amplitude, 12 

Zenith Distance — how measured, 12 

Axis of the earth — axis of the celestial 

sphere, 12 

Poles of the earth — poles of the heav- 
ens, 12 

Equator — terrestrial and celestial, 12 

Hour circles, 13 

Latitude, 13 

Polar Distance, how related to latitude, 13 

Longitude, 13 

Standard Meridians, 13 

Ecliptic, 13 

Inclination of the ecliptic to the equa- 
tor, 13 

Equinoctial points, 13 

Equinoxes — Vernal and autumnal, .... 13 

Solstitial points, 14 

Solstices, 14 

Signs of the ecliptic enumerated, 14 

Colures — Equinoctial and Solstitial,... 14 

Right ascension, 15 

Declination, 15 

Celestial Longitude, 15 

Celestial Latitude, 15 

North Polar Distances, how related to 

latitude, 15 

Parallels of Latitude, 15 

Tropics, 16 

Polar circles, 16 

Zones, 16 

Zodiac, 16 



VI 



ANALYSIS. 



Page. 
Elevation of the pole — to what is it 

equal? 16 

Elevation of the equator, 16 

Distance of a place from the pole, to 

what equal? 16 

Chapter II. — diurnal revolution — arti- 
ficial GLOBES ASTRONOMICAL PROBLEMS. 

Circles of Diurnal Revolution, 17 

Sidereal day defined, 17 

Appearance of the circles of diurnal 

revolution at the equator, 17 

A Right Sphere defined, 18 

A Parallel Sphere, 19 

An Oblique Sphere, 19 

Circle of Perpetual Apparition, 20 

Circle of Perpetual Occupation, 20 

How are the circles of daily motion cut 
by the horizon in the different 

spheres? 20 

Explanation of the peculiar appearan- 
ces of each sphere, from the revolu- 
tion of the earth on its axis, 21 

Artificial Globes — terrestrial and celes- 
tial, 22 

Their use, 23 

Meridian — how represented — how gra- 
duated, 23 

Horizon — how represented — how gra- 
duated, 23 

Hour Circles, how represented, 23 

Hour Index described, 23 

Quadrant of Altitude, 24 

Its use described, 24 

To rectify the globe for any place, 24 

Problems on the terrestrial Globe 
— To find the latitude and longitude 

of a place, 24 

To find a place, its latitude and longi- 

gitude being given, 25 

To find the bearing and distance of 

two places, 25 

To determine the difference of time of 

two places, 25 

The hour being given at any place, to 
tell what hour it is in any other part 

of the world, 25 

To find the anioeci, periceci, and antipo- 
des, 25 

To rectify the globe for the sun's place, 26 
The latitude of the place being given, 
to find the time of the sun's rising 

and setting, 26 

Problems on the celestial Globe. — 
To find the right ascension and decli- 
nation, 26 

To represent the appearance of the 

heavens at any time, 27 

To find the altitude and azimuth of a 
star, 27 



Page. 

To find the angular distance of two 
stars from each other, 27 

To find the sun's meridian altitude, 
the latitude and day of the month 
beinggiven, 28 

Chapter III. — Parallax — Refraction — 
Twilight. 

Parallax defined, 28 

Horizontal Parallax,...^. . <£. ..^ 29 

Relation of parallax to the zenith dis- 
tance, and distance from the center 

of the earth, ..:, ... 29 

To find the horizontal parallax from 

the parallax at any altitude, 29 

Amount of parallax in the zenith and 

in the horizon, 30 

Effect of parallax upon the altitude of 

a body, 30 

Mode of determining the horizontal 

parallax of a body,., 30 

Amount of the sun's nor. par 31 

Use of parallax, 31 

Refraction. — Its effect upon the alti- 
tude of a body, 32 

Its nature illustrated, 32 

Its amount at different angles of eleva- 
tion, 32 

How the amount is ascertained, 33 

Sources of inaccuracy in estimating the 

refraction, 35 

Effect of refraction upon the sun and 

moon when near the horizon, 35 

Oval figure of these bodies explained,. 35 
Apparent enlargement of the sun and 

moon near the horizon, ,. 36 

Twilight. — Its cause explained, 37 

Length of twilight in different latitudes, 37 
How the atmosphere contributes to dif- 
fuse the sun's light, 37 

Chapter IV.— Time. 

Time defined, 38 

What period is a sidereal day, 38 

Uniformity of sidereal days, 38 

Solar time, how reckoned, 39 

Why solar days are longer than side- 
real 39 

Apparent time defined, 39 

Mean' time, 40 

An astronomical day, 40 

Equation of time defined, 40 

When do apparent time and mean 

time differ most? 40 

When do they come together ? 40 

Effect of a change in the place of the 

earth's perihelion, 40 

Causes of the inequality of the solar 

days, 41 

Explain the first cause, depending on 

the unequal velocities of the sun,.... 41 



ANALYSIS. 



Page. 
Explain the second cause, depending 

on the obliquity of the ecliptic, 42 

When does the sidereal day com- 

mence? 44 

The Calendar. — Astronomical year de- 
fined, 45 

How the most ancient nations deter- 
mined the number of days in the year, 45 
Julius Caesar's reformation of the calen- 
dar explained, 45 

Errors of this calendar, 45 

Reformation by Pope Gregory 46 

Rule for the Gregorian calendar, 46 

New style, when adopted in England, 46 
What nations still adhere to the old 

style? 46 

What number of days is now allowed 

between old and new style ? 47 

How the common year begins and ends, 47 

How leap year begins and ends, 47 

Does the confusion of different calen- 
dars affect astronomical observations ? 47 

Chapter V. — Astronomical Instruments 
and Problems — Figure and Density of 
the earth. 

How the most ancient nations acquired 

their knowledge of Astronomy, 48 

Use of instruments in the Alexandrian 

School, 48 

Ditto, by Tycho Brahe, 48 

Ditto, by the Astronomers Royal, 48 

Space occupied by 1" on the limb of an 

instrument, 48 

Extent of actual divisions on the limb, 49 

Vernier, defined, 49 

Its use illustrated, 49 

Chief astronomical instruments enu- 
merated, 50 

Observations taken on the meridian. . . 50 

Reasons of this, 50 

Transit Instrument defined, 51 

Ditto described, 51 

Method of placing it in the meridian. . 51 

Line of collimat ion defined, 52 

System of wires in the focus, 52 

Its use for arcs of right ascension, 52 

Astronomical Clock, — how regulated, 52 

What does it show ? 52 

How to test its accuracy, 53 

How corrected, 53 

Mural Circle, its object, 54 

Describe it, 54 

How the different parts contribute to 

theobject,. 54 

Mural Quadrant, 55 

Use of the Mural Circle for arcs of de- 
clination, 56 

Altitude and Azimuth Instrument de- 
fined, 56 



Page. 

Its use, 56 

Describe it, 57 

Sextant described, 58 

How to measure the angular distance 

of the moon from the sun, 59 

How to take the altitude of a heavenly 

body 59 

Use of the artificial horizon, 59 

In what consists the peculiar value of 

theSextant? 60 

Astronomical Problems. — Given the 
the sun's right ascension and decli- 
nation, to find his longitude and the 

obliquity of the ecliptic, 61 

Napier's Rule of circular parts, 62 

Given the sun's declination to find his 
rising and setting at any place whose 

latitude is known, 63 

Given the latitude of a place and the 
declination of a heavenly body, to 
determine its altitude and azimuth 
when on the six o'clock hour circle, 64 
The latitudes and longitudes of two 
celestial objects being given, to find 

their distance apart, 65 

Figure and Density of the Earth — 
reason for ascertaining it with great 

precision, 66 

How found from the centrifugal force, 66 
From measuring an arc of the meridian, 67 
From observations with the pendulum, 68 

From the motions of the moon, 68 

Density of the earth compared with 

water, 68 

How ascertained by Dr. Maskelyne, . . 69 
Why an important element, 69 

Part II.— OF THE SOLAR SYSTEM. 

Chapter 1. — The Sun — Solar Spots — Zo- 
diacal Light. 

Figure of the sun, 70 

Angle subtended by a line of 400 miles, 70 

Distance from the earth, 70 

Illustrated by motion on a railway car, 70 
Apparent diameter of the sun — how 

found, 72 

How to find the linear diameter, 71 

How much larger is the sun than the 

earth, 71 

Its density and mass compared with 

the earth's, 71 

Weight at the surface of the sun, 72 

Velocity of falling bodies at the sun ... 72 

Solar Spots. — Their number, 72 

Size, 72 

Description, 72 

What region of the sun do they oc- 
cupy, 73 

Proof that they are on the sun, 73 



Vlll 



ANALYSIS. 



Page 
How we learn the revolution of the sun 

on his axis, 73 

Time of the revolution...... 73 

Apparent paths of the spots, 74 

Inclination of the solar axis, 74 

Sun's Nodes — when does the sun pass 

them ? 75 

Cause of the solar spots, 76 

Facula?, 76 

Zodiacal Light. — Where seen, 76 

Its form, 76 

Aspects at different seasons, 76 

Its motions, 77 

Its nature, 77 

Chapter II. — Apparent Annual Motion 

of the Sun — Seasons — Figure of the 
Earth's Orbit. 

Apparent motion of the sun, 78 

How both the sun and earth are said to 

move from west to east, 79 

Nature and position of the sun's orbit, 

how determined, 79 

Changes in declination how found, 79 

Ditto, in right ascension, 80 

Inferences from a table of the sun's de- 
clinations, 80 

Ditto, of right ascensions, 81 

Path of the sun, how proved to be a 

great circle, 81 

Obliquity of the ecliptic, how found, 81 

How it varies, 81 

Great dimensions of the earth's orbit, 81 

Earth's daily motion in miles, 82 

Ditto, hourly ditto, 82 

Diurnal motion at the equator per 

hour, 82 

Seasons. — Causes of the change of sea- 
sons, 82 

How each cause operates, 82 

Illustrated by a diagram, 83 

Change of seasons had tbe equator been 

perpendicular to the ecliptic, 84 

Figure of the Earth's Orbit. — Proof 

that the earth's orbit is not circular, 85 

Radius vector defined, 85 

Figure of the earth's orbit how ob- 
tained, 86 

Relative distances of the earth from the 

sun, how found, 86 

Perihelion and Aphelion defined, 87 

Variations in the sun's apparent diame- 
ter, 87 

Angular velocities of the sun at the pe- 
rihelion and aphelion, 87 

Ratio of these velocities to the dis- 
tances, 87 

How to calculate the relative distances 
of the earth from the sun's daily mo- 
tions, 88 



Page 
Product of the angle described in any 
given time by the square of the dis- 
tance, 88 

Space described by the radius vector of 

the solar orbit in equal times, 88 

How to represent the sun's orbit by a 
diagram, 89 

Chapter III. — Universal Gravitation. 

Universal Gravitation defined, 90 

Why is it called attraction, 90 

History of its discovery, 90 

How was the gravitation of the moon 

to the earth first inferred? 91 

Laws of Gravitation. — If a body re- 
volves about an immovable center 
of force, and is constantry attracted 

to it, how will it move? 92 

If a body describes a curve around a 
center towards which it tends by any 
force, how is its angular velocity re- 
lated to the distance, 93 

In the same curve, the velocity at any 

point of the curve varies as what ? 93 
If equal areas be described about a cen- 
ter in equal times, to what must the 

force tend? 94 

How is the distance of any planet from 
the sun at any point in its orbit, to 
its distance from the superior focus ? 94 
Case of two bodies gravitating to the 
same center where one descends in a 
straight line, and the other revolves 

in a curve, 95 

Velocity of a body at any point when 

falling directly to the sun, 97 

Relation between the distances and pe- 
riodic times, 99 

Kepler 's three great laws, 99 

Motion in an Elliptical Orbit, 100 

Idea of a projectile force, 100 

Nature of the impulse originally given 

to the earth, 100 

Two forces under which a body re- 
volves, 100 

Illustrated by the motion of a cannon 

ball, 101 

Why a planet returns to the sun, 102 

Illustration by a suspended ball, 103 

Chapter IV. — Precession of the Equi- 
noxes — Nutation — Aberration — Mean 
and True Places of the sun. 

Precession of the Equinoxes defined, 104 

Why so called, 104 

Amount of Precession annually, 104 

Revolution of the equinoxes, 104 

Revolution of the pole of the equator 
around the pole of the ecliptic, 105 



ANALYSIS. 



IX 



Page. 
Changes among the stars caused by 

precession, 105 

The present pole star not always such, 105 
What will be the pole star 13,000 

years hence ? 105 

Cause of the precession of the equi- 
noxes, 105 

Explain how the cause operates, 106 

Proportionate effect of the sun and 

moon in producing precession, 107 

Tropical year defined, 107 

How much shorter than the sidereal 

year, 107 

Use of the precession of the equinoxes 

in chronology, 107 

Nutation, defined, 108 

Explain its operation, 106 

Cause of Nutation, 106 

Aberration, defined, 108 

Illustrated by a diagram, 10 ( J 

Amount of aberration, 109 

Effect on the places of the stars, 10!) 

Motion of the Apsides, the fact sta- 
ted, 109 

Direction of this motion, 110 

Time of revolution of the line of Ap- 
sides, 110 

Present longitude of the perihelion,. . 110 

When was it nothing? 110 

Mean and True Places of the Sun, 111 

Mean Motion defined, Ill 

Illustrated by surveying a field, Ill 

Mean and true longitude distinguish- 
ed, Ill 

Equations defined, Ill 

Their object, Ill 

Mean and True anomaly defined,.... 112 

Equation of the Center, 112 

Explain from the figure, 112 



Chapter V. — The Moon — Lunar Geogra- 
j'fiv — Phases of the Moon — Her Revo- 
lutions. 

Distance of the moon from the earth, 113 

Her mean horizontal parallax, 113 

Her diameter, 113 

Volume, density, and mass, 113 

Shines by reflected light, 113 

Appearance in the telescope, 113 

Terminator defined, 113 

Its appearance, 113 

Proofs of Valleys, 114 

Form of these, 114 

Best time for observing the lunar 

mountains and valleys, 114 

Names of places on the moon double, 115 

Dusky regions how named, 115 

Point out remarkable places on the 

map of the moon, 115 

Explain the method of estimating the 

height of lunar mountains, 115 

B 



Specify the heights of particular 
mountains, 

Volcanoes, proof of their existence, .... 

Has the moon an atmosphere ? 

Improbability of identifying artificial 
structures in the moon, 

Phases of the Moon, their cause,.... 

Successive appearances of the moon 
from one new moon to another,.... 

Syzygies defined, 

Explain the phases of the moon from 
figure 46, 

Revolutions of the moon. Period 
of her revolutions about the earth,. 

Her apparent orbit a great circle, 

A sidereal month defined, 

A synodical do. 

Length of each, 

Why the synodical is longer, 

How each is obtained, 

Inclination of the lunar orbit, 

Nodes defined, 

Why the moon sometimes runs high 
and sometimes low, 

Harvest moon defined, 

Ditto explained, 

Explain why the moon is nearer to us 
when on the meridian than when 
near the horizon, 

Time of the moon's revolution on its 
axis, 

How known, 

Librations explained, 

Diurnal Libration, 

Length of the Lunar days, 

Earth never seen on the opposite side 
of the moon, 

Appearances of the earth to a specta- 
tor on the moon, 

Why the earth would appear to re- 
main fixed, 

Ascending and descending nodes dis- 
tinguished, 

Whether the earth carries the moon 
around the sun, 

How much more is the moon attract- 
ed towards the sun than towards 
the earth, 

When does the sun act as a disturbing 
force upon the moon ? 

Why docs not the moon abandon the 
earth at the conjunction ? 

The moon's orbit concave towards the 



sun, 



How the elliptical motion of the moon 
about the earth is to be conceived 
of, 

Illustrations, 



Page. 

117 
117 
117 

117 
118 

118 
118 

119 

119 
120 
120 
120 
120 
120 
120 
121 
121 

121 
122 
122 



122 

123 
123 

123 
12 1 
124 

124 

124 

125- 

125 

126 

126 
126 
126 
127 



127 

127 



Chapter VI. — Lunar Irregularities. 
Specify their general cause, 127 



ANALYSIS. 



Page. 
Unequal action of the sun upon the 

earth and moon, 128 

Oblique action of earth and sun, 128 

Gravity of the moon towards the 

earth at the syzygies, 129 

Gravity at the quadratures, 129 

Explain the disturbances in the 

moon's motions from figure 48, 130 

Figure of the moon's orbit, 132 

How its figure is ascertained, 132 

Moon's greatest and least apparent di- 
ameters, 132 

Her greatest and least distances from 132 

the earth, 132 

Perigee and Apogee defined, 132 

Eccentricities of the solar and lunar 

orbits compared, 133 

Moon's nodes, their change of place,. 133 

Rate of this change per annum, , 133 

Period of their revolution, 133 

Irregular curve described by the 

moon, 133 

Cause of the retrograde motion of 

nodes, 133 

Explain from figure 50, 134 

Synodical revolution of the node de- 
fined, 135 

Its period, 135 

The Saros explained, 1 35 

The Metonic Cycle,. 135 

Golden Number, 136 

Revolution of the line of apsides, 136 

Its period, 136 

How the places of the perigee may be 

found, 136 

Moon's anomaly defined, 136 

Cause of the revolution of the apsides, 136 

Amount of the equation of the Center, 13 

Evection defined, 137 

Its cause explained, 138 

Variation defined, 140 

Its cause, 140 

Annual Equation explained, 140 

How these irregularities were first 

discovered, 141 

How many equations are applied to 

the moon's motions ? 141 

Method of proceeding in finding the 

moon's place, 141 

Successive degrees of accuracy at- 
tained, 141 

Periodic and secular irregularities dis- 
tinguished, 141 

Acceleration of the moon's mean mo- 
tion explained, 141 

Its consequences, 142 

Lunar inequalities of latitude and 

parallax, 142 

Chapter VII. — Eclipses. 

Eclipse of the moon, when it happens, 143 



Eclipse of the sun, when it happens, . 143 

When only can each occur, 143 

Why an eclipse does not occur at 

every new and full moon, 144 

Why eclipses happen at two opposite 

months, 144 

Circumstances which affect the length 

of the earth's shadow, 144 

Semi-angle of the cone of the earth's 

shadow, to what equal, 145 

Length of the earth's shadow, 145 

Its breadth where it eclipses the 

moon, 146 

Lunar ecliptic limit defined, 146 

Solar, ditto 146 

Amount of the lunar ecliptic limit,.... 146 

Appulse defined, 147 

Partial, total, central, eclipse, each 

defined, 147 

Penumbra defined, 147 

Semi-angle of the moon's penumbra, 

to what equal, 148 

Semi-angle of a section of the penum- 
bra where the moon crosses it, 148 

Moon's horizontal parallax increased 

eV wh y. • 148 

Why the moon is visible in a total 

eclipse, 148 

Calculation of eclipses, general mode 

of proceeding, 149 

To find the exact time of the begin- 
ning, end, duration, and magnitude 
of a lunar eclipse, by figures 53, 54, 150 

Elements of an eclipse defined, ..» 151 

Digits defined, 153 

How the shadow of the moon travels 

over the earth in a solar eclipse,.... 153 
Why the calculation of a solar eclipse 
is more complicated than a lunar,. 154 

Velocity of the moon's shadow, 154 

Different ways in which the shadow 
traverses the earth, according as 
the conjunction is near the node or 

near the limit, 155 

When do the greatest eclipses hap- 
pen ? 155 

Case in which the moon's shadow 

nearly reaches the earth, 156 

How far may the shadow reach be- 
yond the center of the earth ? 157 

Greatest diameter of the moon's sha- 
dow where it traverses the earth,.. 157 
Greatest portion of the earth's surface 
ever covered by the moon's penum- 
bra, 157 

Moon's apparent diameter compared 

with the sun's, 158 

Annular eclipse, its cause, 158 

Direction in which the eclipse passes 
on the sun's disk, 159 



ANALYSIS. 



XI 



Page. 

Greatest duration of total darkness,. . . 159 

Eclipses of the sun more frequent 
than of the moon, why ? 159 

Lunar eclipses oftener visible, why ? 159 

Radiation of light in a total eclipse of 
the sun, 160 

Interesting phenomena of a total 
eclipse of the sun, 160 

Phenomena of the eclipse of 1 806, de- 
scribed, 160 

When does the next total eclipse of 
the sun, visible in the United 
States, occur? 161 

Chapter VIII. — Longitude. — Tides. 
Objects of the ancients in studying 

astronomy, 161 

Ditto of the moderns, 161 

Longitude. — How to find the differ- 
ence of longitude between two 

places, 161 

Method by the Chronometer explain- 
ed, 162 

How to set the chronometer to Green- 
wich time, 162 

Accuracy of some chronometers, 162 

Objections to them, 162 

Longitude by eclipses explained, 163 

Lunar method of finding the longi- 
tude, 163 

Circumstances which render this 

method somewhat difficult, 164 

Disadvantages of this method," 164 

Degree of accuracy attainable, 165 

Tides. — defined, 1 65 

High, Low, Spring, Neap, Flood, and 

Ebb Tide, severally defined, 165 

Similar tides on opposite sides of the 

earth, 165 

Interval between two successive high 

tides, 165 

Average height for the whole globe, 166 

Extreme height, 166 

Cause of the tides, 1 66 

Explain by figure 56, 166 

Tide-wave defined, 167 

Comparative effects of the sun and 

moon in raising the tide, 167 

Why the moon raises a higher tide 

than the sun, 167 

Spring tides accounted for, 168 

Neap tides, ditto 168 

Tower of the sun or moon to raise 
the tide, in what ratio to its dis- 

tance, 168 

Influence of the declinations of the 

sun and moon on the tides, 169 

Explain from figures 57 and 58, 169 

Motion of the tide-wave not progres- 
sive, 170 

Tides of rivers, narrow bays, how 
produced, 170 



Page. 

Cotidal Lines defined, 170 

Derivative and Primitive tides distin- 
guished, 170 

Velocity of the tide- wave, circum- 
stances which affect it, 171 

Explain by figure 59, 171 

Examples of very high tides, 172 

Unit of altitude defined, 172 

Unit of altitude for different places, 172 
Tides on the coast of N. America, 

whence derived, 173 

Why no tides in lakes and seas, 173 

Intricacy of the problem of the tides, 173 

Atmospheric tide, 173 

Chapter IX. — Planets — Inferior Plan- 
ets, Mercury and Venus. 

Etymology of the word planet, 174 

Planets known from antiquity, 174 

Ditto, recently discovered, 174 

Primary and Secondary Planets dis- 
tinguished, 174 

Whole number of each, 175 

Inclination of their orbits, 175 

Inferior and superior planets distin- 
guished, 175 

Differences among the planets, 175 

Distances from the sun in miles, 175 

Great dimensions of the planetary 

orbits, 176 

How long a railway car would re- 
quire to cross the orbit of Uranus, 176 

Law of the distances, 176 

Mean distances, how determined, — 176 

Din meters in miles, 177 

How ascertained, 177 

Periodic limes, 178 

Which planets move most rapidly,... 178 
Inferior Planets. — Their proximity 

to the sun, 178 

Illustrate by figure 60, 179 

When is a planet said to be in con- 
junction, 179 

Interior and superior conjunctions 

distinguished, 179 

Synodical revolution of a planet de- 
fined, 179 

Why its period exceeds that of the 

planet in its orbit, 179 

To ascertain the synodical period 

from the sidereal, 180 

Synodical periods of Mercury and 

'Venus, 180 

Motions of an inferior planet de- 
scribed, 180 

Explain from figure 60, 180 

When is an inferior planet station- 
ary, 181 

Elongation of Mercury — when sta- 
tionary, 1 81 

Ditto of Venus, 181 

Phases of Mercury and Venus, 182 



XII 



ANALYSIS. 



Page.! 

Distance of an inferior planet from 

the sun, how found, 182 

Eccentricity of the orbit of Mercury, 182 

Ditto of Venus, 182 

Most favorable time for determining 

the sidereal revolution of a planet, 183 

"When is an inferior planet brightest, 183 

Times of their revolutions on their 

axes, 183 

Venus, her brightness, 183 

Her conjunctions with the sun every 

eight years, 184 

Transits of the Inferior Planets 

defined, 184 

Why a transit does not occur at 

every inferior conjunction, 184 

Why the transits of Mercury occur in 
May and November, and those of 

Venus in June and December, 185 

Intervals between successive transits, 

how found, 185 

Why transits of Venus sometimes oc- 
cur after an interval of eight years, 186 

Why transits are objects of so much 

interest, 186 

Method of finding the sun's horizontal 

parallax from the transit of Venus, 187 

Why distant places are selected for 

observing it, 187 

Explain the principle from figure 61, 187 

Amount of the sun's horizontal par- 
allax, 188 

Indications of an atmosphere in Venus, 189 

Whether Venus has any satellite, — 189 

Mountains of Venus, 189 

Chapter X. — Superior Planets, Mars, Ju- 
piter, Saturn, and Uranus. 

How the superior planets are distin- 
guished from the inferior, 189 

Mars, diameter, 190 

Mean distance from the sun, 190 

Inclination of his orbit, 190 

Variation of brightness and magnitude, 190 

Explain the cause from figure 62, . ... 190 

Phases of Mars, 191 

Telescopic appearance, 191 

Revolution on his axis, 191 

Spheroidal figure, 191 

Jupiter — great size, diameter, 191 

Spheroidal figure, 192 

Rapid diurnal revolution...... 192 

Distance from the sun, 192 

Periodic time, 192 

Telescopic appearance, 192 

Why astronomers regard Jupiter and 

his moons with so much interest,... 192 

Belts, number, situation, cause, 192 

Satellites of Jupiter, number, situa- 
tion, ,.., , 193 



Page. 

Motions of the satellites, 193 

Diameter, 193 

Distances from the primary, 193 

Figure of their orbits, 194 

Their inclination to the planet's equa- 
tor, 194 

Their eclipses, how they differ from 

from those of the moon, 194 

Their phenomena explained from fig- 
ure 63, 195 

Shadow seen traversing the disk of 

the primary, 196 

Satellite itself seen on the disk, 196 

Remarkable relation between the 
mean motions of the three first 

satellites, 196 

Consequences of this, 196 

Use of the eclipses of Jupiter's satel- 
lites in finding the longitude, 197 

How it is adapted to this purpose,... 197 

Imperfections of this method, 197 

Why not practised at sea, 198 

Discovery of the progressive motion 

of light, how made, 198 

Saturn, size, 199 

Number of satellites, 199 

Ring, double, 199 

Dimensions of the ring in several par- 
ticulars, 199 

Representation in the figure, (frontis- 
piece,) 199 

Proof that the ring is solid and opake, 199 
Proof that the axis of rotation is per- 
pendicular to the plane of the ring, 199 

Period of rotation, 200 

Compression of the poles, 200 

Peculiar figure of the planet, 200 

Parallelism of the ring in all parts of 

its revolution, 200 

Illustration by a small disk and a 

lamp, 200 

Different appearances of the ring, .... 200 

Explain diagram 64, 201 

Proof that the ring shines by reflected 

light, 202 

Revolution of the ring, 202 

How ascertained, 202 

Rings not concentric with the planet, 203 

Advantages of this arrangement, 203 

Appearance of the rings from the 

planet, 203 

Satellites, distance of the outermost 

from the planet, 204 

Description of the satellites, 204 

Uranus, distance and diameter, 204 

Period of revolution, inclination of its 

orbit, 205 

Appearance of the sun from Uranus, 205 

History of its discovery, 205 

Satellites, number, minuteness, 205 

Irregular motions, 205 



ANALYSIS. 



Xlll 



Page 

New Planets, their names, 206 

Position in the system, 206 

Discovery, 206 

Theoretic notions respecting their ori- 
gin, 206 

Reason of their names, 20 

Their average distance, 207 

Periodic Times, 207 

Inclinations of their orbits, 207 

Eccentricity of do. 207 

Small size, 206 

Atmospheres, 208 

Chapter XI. — Motions of the Planetary 
System. 

Reasons for delaying the consideration 
of the planetary motions, 208 

Two methods of studying the celestial 
motions, 208 

Notions of absolute space 209 

Appearance of the planets from the 
sun, 209 

Particular appearance of the orbit of 
Mercury, 210 

Mutual relation of the orbit of the 
earth and Mercury considered, 210 

How the motions of the other plan- 
ets differ from from those of Mer- 
cury, 210 

Why is the ecliptic taken as the stan- 
dard of reference, 210 

Three particulars necessary in order 
to represent the actual positions of 
the planetary orbits, 211 

\\ liy diagrams represent the orbits er- 
roneously, as figure 65, 211 

Inadequate representations of the so- 
lar system, 213 

How the planets would be truly repre- 
sented, 213 

Two reasons why the apparent mo- 
tions are unlike the real, 213 

Explain figure 66, 214 

Motions of Venus compared with those 
of Mercury, 215 

Apparent motions of the superior plan- 
ets, how far they are like and how 
unlike the inferior, 215 

Explain figure 67, 215 

Chapter XII. — Determination ofthe Plan- 
etary Orbits — Kepler's Discoveries — 
Elements of the Orbit of a Planet — 
Quantity of Matter in the Sun and 
Planets — Stability of the Solar Sys- 
tem. 

Ptolemy's views of the figure of the 
planetary orbits, 217 

Kepler's investigation of the motions 
of Mars, 218 



Page. 

History of the discovery of Kepler's 
Laws, 218 

Third Law, how modified by the quan- 
tity of matter, 219 

Elements, their number, 220 

Enumeration of them, 220 

Why we cannot find them as we do 
those of the moon and sun, 220 

First steps in the process, 221 

To find the heliocentric longitude and 
latitude of a planet, figure 68, 221 

To find the position of the nodes, and 

the inclination, figure 69, 222 

To find the periodic time, 223 

Difficulty of finding when a planet is 
at its node, 223 

Advantage of observations taken when 
a planet is in opposition, 223 

Periodic time, how ascertained most 
accurately, 224 

To find the major axis of the planet- 
ary orbits, 224 

Constancy of the major axis, 225 

To find the place of the perihelion, 
figure 71, 226 

To find the place of the planet in its 
orbit at a particular epoch, 227 

To find the eccentricity, 227 

Quantity of Matter in the Sun 
and Planets. — How we learn the 
quantity of matter in a body, 228 

Method by means of the distances and 
periodic times of their satellites, 228 

Mass of the sun compared with that of 
the earth , 229 

Same result how deduced from the 
centrifugal force, 229 

Mass of the planets that have no sat- 
ellites how found, 229 

How the quantity of matter in bodies 

varies 230 

How their densities vary, 230 

Inferences from the table of densities 
and specific gravities ofthe planets, 230 

Perturbations produced by the planets, 231 

Stability of the Solar System. — 
Probability of derangement in the 
planetary motions, 231 

Actual changes, 231 

Result of the investigations of La Place 
and La Grange, 232 

Important relation between great 
masses and small eccentricities,... . 233 

Chapter XIII. — Comets. 

Three parts of a comet 234 

Description of each part, 234 

Number of Comets, 234 

Six particularly remarkable 235 

Differences in magnitude and bright- 
ness, 236 



XIV 



ANALYSIS. 



Page 
Variations in the same comet at dif- 
ferent returns, 236 

Periods of comets, 237 

Distances of their aphelia, 237 

Proof that they shine by reflected 

light, 237 

Changes in the tail at different dis- 
tances from the sun, 237 

Direction from the sun, 237 

Quantity of Matter, 238 

Effect when they pass very near the 

planets, 238 

Proof that they consist of matter, 238 

How a comet's orbit may be entirely 

changed, 239 

How exemplified in the comet of 1770, 239 
Orbits and Motions of Comets. — 

Nature of their Orbits, 240 

Five Elements of a Comet, 240 

Investigation of these elements, why 

so difficult, 241 

Can the length of the major axis be 

calculated? 242 

How determined, 242 

Elements of a comet, how calculated 243 
How a comet is known to be the same 

as one that has appeared before, 243 

Exemplified in Halley's comet, 243 

Return in 1835, 244 

Encke's comet, appearance in 1839, 244 

Proofs of a Resisting Medium, 244 

Its consequences, 244 

Physical nature of comets, 245 

How their tails are supposed to be 

formed, 245 

Difficulty of accounting for the direc- 
tion of the tail, 245 

Supposition of Delambre, 246 

Possibility of a comet's striking the 

earth, 246 

Instances of comets coming near the 

earth, 247 

Consequences of a collision, 247 

Part III.— OF THE FIXED STARS 
AND SYSTEM OF THE WORLD. 

Chapter I. — Fixed Stars — Constellations. 

Fixed stars, why so called, 248 

Magnitudes, how many visible to the 

naked eye, 248 

Whole number of magnitudes, 248 

Antiquity of the constellations, 248 

Whether the names are founded on 

resemblance, 249 

Names of the individual stars of a con- 
stellation, 249 

Catalogues of the stars, 249 

Numbers in different catalogues, 249 

Utility of learning the constellations, 250 



Page. 
Constellations. — Aries, how recog- 
nized, 251 

Taurus do. — largest star in Taurus,... 251 
Gemini, — magnitude of Castor, of Pol- 
lux, 251 

Cancer, size of its stars, Prsesepe, . . . . 251 
Leo, size, magnitude of Regulus, sickle, 

Denebola, 251 

Virgo, direction, Spica, Vindemiatrix, 252 

Libra, how distinguished, 252 

Scorpio, his head how formed, An- 

tares, tail, 252 

Sagittarius, direction from Scorpio, 

how recognized, 252 

Capricornus, direction from Sagitta- 
rius, two stars, 

Aquarius, its shoulders, 252 

Pisces, situation, 252 

Piscis Australis, Fomalhaut, 252 

Andromeda, how characterized, Mi- 

rach, Almaak, 253 

Perseus, Algol, Algenib, 253 

Auriga, situation, — Capella, its mag- 
nitude, 253 

Lynx,... 253 

Leo Minor, situation from Leo, 253 

Coma Berenices, direction from Leo, 

CorCaroli, 253 

Bootes, Arcturus, size and color, 253 

Corona Borealis, where from Bootes, 

figure, 254 

Hercules, number of stars, great extent, 254 

Ophiuchus, where from Hercules,.... 254 

Aquila, three stars, Altair, Antinous, 254 

Delphinus, four stars, tail, 254 

Pegasus, four stars in a square, their 

names, 254 

Ursa Minor, Pole-star, Dipper, 254 

Ursa Major, how recognized, Point- 
ers, Alioth, Mizar, 255 

Draco, position with respect to the 

Great and Little Bear, 255 

Cepheus, where from the Dragon, size 

of its stars, 255 

Cassiopeia's chair — in the Milky Way, 255 
Cygnus, where from Cassiopeia, fig- 
ure, 255 

Lyra, largest star, 255 

Cetus, its extent, Menkar, Mira, 256 

Orion, size and beauty, parts, 256 

Canis Major, where from Orion, Sirius, 256 

Canis Minor, Procyon, 256 

Hydra — situation — Cor Hydras, 256 

Corvus, how represented, 256 

Chapter II. — Clusters of Stars — Nebula 
— Variable Stars — Temporary Stars — 
Double Stars. 

Clusters. — Examples, 257 

Number of stars in the Pleiades, 257 



ANALYSIS. 



XV 



Page 
Stars of Coma Berenices and Prse- 

sepe, 257 

Nebula. — Defined, 258 

Examples, 258 

Number in Herschel's Catalogue, .... 258 

Herschel's Views of their nature, 259 

Figures of nebula?, 259 

Nebula in the Sword of Orion, 259 

Nebulous Stars, defined, figures, 259 

Annular Nebula, appearance, exam- 

pie, 260 

Galaxy or Milky Way, Herschel's 

views of it, 260 

Variable Stars. — Defined, 260 

Examples in o Ceti and Algol, 261 

Temporary Stars. — Defined, 261 

Examples, why Hipparchus number- 
ed the stars, 261 

Stars seen by the ancients, now miss- 
ing, 261 

Double Stars. — Defined, examples,. 262 
Distance between the double stars in 

seconds, 262 

Colors of sonic double stars, 262 

Examnles, 263 

Number, 263 



Chapter III. — Motions of the Fixed 
Stars — Distances — Nature. 

Binary Stars, how distinguished 
from common double stars, 264 

Examples of revolving si ars, 265 

Inferences from the tabular view,.... 265 

Particulars of y Virginis, 265 

Proof that the law of gravitation ex- 
tends to the stars, 266 

Whether these stars are of a planeta- 
ry or a cometary nature, 267 

Proper Motions. — Result on com- 
paring the places of certain stars 
with those they had in the time of 
Ptolemy, 267 

Conclusion respecting the apparent 
motions of certain stars, 267 

How the fact of the sun's motion 
might be proved, 267 

Example of stars having a proper mo- 
tion, 267 

What class of stars have the greatest 
proper motion? 268 



Page. 
Distances of the Fixed Stars. — 

What we can determine respecting 

the distance of the nearest star,.... 268 

Base line for measuring this distance, 269 

Have the stars any parallax ? 269 

Taking the parallax at 1", find the 

distance, 269 

Amount of this distance, 269 

Probable greater distance of the 

smaller stars, 270 

Disputes respecting the parallax of 

the stars, 270 

To find the parallax by means of the 

double stars, 270 

Minuteness of the angles estimated, . 270 
How the magnitude of the stars is 

affected by the telescope, 271 

Nature of the Stars. — Magnitude 

compared with the earth, 271 

Dr. Wollaston's observation on their 

comparative light, 271 

Proofs that they are suns, 272 

Arguments for a Plurality of Worlds, 272 

Chapter IV. — Of the System of the 
World. 

System of the world defined, 273 

Compared to a machine, 273 

Astronomical knowledge of the an- 
cients, 373 

Things known to Pythagoras, 273 

His views of the system of the world, 273 

Cyrstalline Spheres of Eudoxus, 274 

Knowledge possessed by Hipparchus, 275 

Ptolemaic Syste?n, 276 

Deferents and epicycles defined, 276 

Explained by figure 74, 276 

How far this system would explain 

the phenomena, 277 

Us absurdities, 277 

Objections to the Ptolemaic System,. 278 

Tychonic System, 278 

Its advantages, 278 

Its absurdities, 278 

Copemican System, 279 

Proofs that the earth revolves, 279 

Ditto, that the planets revolve about 

the sun, 279 

Higher orders of relations among the 

stars, 280 

Proofs of such orders, 280 

Structure of the material universe... 281 



[O 3 Diagrams for public recitations. 



As many of the figures of this work are too complicated to be 
drawn on the black-board at each recitation, we have found it 
very convenient to provide a set of permanent cards of paste- 
board, on which the diagrams are inscribed on so large a scale, as 
to be distinctly visible in all parts of the lecture room. The let- 
ters may be either made with a pen, or better procured of the 
printer, and pasted on. 

The cards are made by the bookbinder, and consist of a thick 
paper board about 18 by 14 inches, on each side of which a white 
sheet is pasted, with a neat finish around the edges. A loop at- 
tached to the top is convenient for hanging the card on a nail. 

Cards of this description, containing diagrams for the whole 
course of mathematical and philosophical recitations, have been 
provided in Yale College, and are found a valuable part of our ap- 
paratus of instruction. 



INTRODUCTION TO ASTRONOMY. 



PRELIMINARY OBSERVATIONS. 



1. Astronomy is that science which treats of the heavenly bodies. 

More particularly, its object is to teach what is known respect- 
ing the Sun, Moon, Planets, Comets, and Fixed Stars ; and also to 
explain the methods by which this knowledge is acquired. Astron- 
omy is sometimes divided into Descriptive, Physical, and Practi- 
cal. Descriptive Astronomy respects facts ; Physical Astronomy. 
causes; Practical Astronomy, the means of investigating the facts, 
whether by instruments, or by calculation. It is the province of 
Descriptive Astronomy to observe, classify, and record, all the 
phenomena of the heavenly bodies, whether pertaining to those 
bodies individually, or resulting from their motions and mutual 
relations. It is the part of Physical Astronomy to explain the 
causes of these phenomena, by investigating and applying the 
general laws on which they depend ; especially by tracing out all 
the consequences of the law of universal gravitation. Practical 
Astronomy lends its aid to both the other departments. 

2. Astronomy is the most ancient of all the sciences. At a 
period of very high antiquity, it was cultivated in Egypt, in Chal- 
dea, in China, and in India. Such knowledge of the heavenly 
bodies as could be acquired by close and long continued observa- 
tion, without the aid of instruments, was diligently amassed ; and 
tables of the celestial motions were constructed, which could be 
used in predicting eclipses, and other astronomical phenomena. 

About 500 years before the Christian era, Pythagoras, of 
Greece, taught astronomy at the celebrated school at Crotona, and 
exhibited more correct views of the nature of the celestial mo- 
tions, than were entertained by any other astronomer of the an- 
cient world. His views, however, were not generally adopted, 

1 



2 , PRELIMINARY OBSERVATIONS. 

but lay neglected for nearly 2000 years, when they were revived 
and established by Copernicus and Galileo. The most celebrated 
astronomical school of antiquity was at Alexandria, in Egypt, 
which was established and sustained by the Ptolemies, (Egyptian 
princes,) about 300 years before the Christian era. The employ- 
ment of instruments for measuring angles, and the introduction of 
trigonometrical calculations to aid the naked powers of observa- 
tion, gave to the Alexandrian astronomers great advantages over 
all their predecessors. The most able astronomer of the Alexan- 
drian school was Hipparchus, who was distinguished above all the 
ancients for the accuracy of his astronomical measurements and 
determinations. The knowledge of astronomy possessed by the 
Alexandrian school, and recorded in the Almagest, or great work 
of Ptolemy, constituted the chief of what was known of our 
science during the middle ages, until the fifteenth and sixteenth 
centuries, when the labors of Copernicus of Prussia, Tycho Brake 
of Denmark, Kepler of Germany, and Galileo of Italy, laid the 
solid foundations of modern astronomy. Copernicus expounded 
the true theory of the celestial motions ; Tycho Brahe carried 
the use of instruments and the art of astronomical observation to 
a far higher degree of accuracy than had ever been done before ; 
Kepler discovered the great laws of the planetary motions ; and 
Galileo, having first enjoyed the aid of the telescope, made innu- 
merable discoveries in the solar system. Near the beginning of 
the eighteenth century, Sir Isaac Newton discovered, in the law 
of universal gravitation, the great principle that governs the ce- 
lestial motions ; and recently, La Place has more fully completed 
what Newton began, having followed out all the consequences of 
the law of universal gravitation, in his great work, the Mecan- 
ique Celeste. 

3. Among the ancients, astronomy was studied chiefly as sub- 
sidiary to astrology. Astrology was the art of divining future 
events by the stars. It was of two kinds, natural and judicial. 
Natural Astrology, aimed at predicting remarkable occurrences in 
the natural world, as earthquakes, volcanoes, tempests, and pesti- 
lential diseases. Judicial Astrology, aimed at foretelling the fates 
of individuals, or of empires. 



PRELIMINARY OBSERVATIONS. 3 

4. Astronomers of every age, have been distinguished for their 
persevering industry, and their great love of accuracy. They 
have uniformly aspired to an exactness in their inquiries, far be- 
yond what is aimed at in most geographical investigations, satis- 
fied with nothing short of numerical accuracy, wherever this is 
attainable ; and years of toilsome observation, or laborious calcu- 
lation, have been spent with the hope of attaining a few seconds 
nearer to the truth. Moreover, a severe but delightful labor is 
imposed on all who would arrive at a clear and satisfactory knowl- 
edge of the subject of astronomy. Diagrams, artificial globes, 
orreries, and familiar comparisons and illustrations, proposed by 
the author or the instructor, may afford essential aid to the learner, 
but nothing can convey to him a perfect comprehension of the 
celestial motions, without much diligent study and reflection. 

5. In expounding the doctrines of astronomy, we do not, as in 
geometry, claim that every thing shall be proved as soon as as- 
serted. We may first put the learner in possession of the leading 
facts of the science, and afterwards explain to him the methods 
by which those facts were discovered, and by which they may 
be verified ; we may assume the principles of the true system of 
the world, and employ those principles in the explanation of many 
subordinate phenomena, while we reserve the discussion of the 
merits of the system itself, until the learner is extensively ac- 
quainted with astronomical facts, and therefore better able to ap- 
preciate the evidence by which the system is established. 

6. The Copernican System is that which is held to be the true 
system of the world. It maintains (1,) That the apparent diur- 
nal revolution of the heavenly bodies, from east to west, is owing 
to the real revolution of the earth on its own axis from west to 
east, in the same time ; and (2,) That the sun is the center around 
which the earth and planets all revolve from w T est to east, con- 
trary to the opinion that the earth is the center of motion of the 
sun and planets. 

7. We shall treat, first, of the Earth in its astronomical rela- 
tions ; secondly, of the Solar System ; and, thirdly, of the Fixed 
Stars. 



PART I. OF THE EARTH. 



CHAPTER I. 

OF THE FIGURE AND DIMENSIONS OF THE EARTH, AND THE DOCTRINE 
OF THE SPHERE. 



8. The figure of the earth is nearly globular. This fact is 
known, first, by the circular form of its shadow cast upon the 
moon in a lunar eclipse ; secondly, from analogy, each of the 
other planets being seen to be spherical ; thirdly, by our seeing 
the tops of distant objects while the other parts are invisible, as 
the topmast of a ship, while either leaving or approaching the 
shore, or the lantern of a light-house, which, when first descried 
at a distance at sea, appears to glimmer upon the very surface of 
the water ; fourthly, by the depression or dip of the horizon when 
the spectator is on an eminence ; and, finally, by actual observa- 
tions and measurements, made for the express purpose of ascer- 
taining the figure of the earth, by means of which astronomers are 
enabled to compute the distances from 

the center of the earth of various places Flg * ** 

on its surface, which distances are found SS5 
to be nearly equal. 

9. The Dip of the Horizon, is the ap- 
parent angular depression of the hori- 
zon, to a spectator elevated above the 
general level of the earth. The eye 
thus situated, takes in more than a ce- 
lestial hemisphere, the excess being the 
measure of the dip. 

Thus, in Fig. 1, let AO represent the 




FIGURE AND DIMENSIONS. 5 

height of a mountain, ZO the direction of the plumb line, HOR a 
line touching the earth at the point O, and at right angles to the 
plumb line, C the center of the earth, DAE the portion of the 
earth's surface seen from O ; OD, OE, lines drawn from the 
place of the spectator to the most distant parts of the horizon, 
and CD a radius of the earth. The dip of the horizon is the an- 
gle HOD or ROE. Now the angle made between the direction 
of the plumb line and that of the extreme line of the horizon or 
the surface of the sea, namely, the angle ZOD, can be easily 
measured ; and subtracting the right angle ZOH from ZOD, the 
remainder is the dip of the horizon, from which the length of the 
line OD may be calculated, (see Art. 10,) the height of the spec- 
tator, that is, the line OA, being known. This length, to whatever 
point of the horizon the line is drawn, is always found to be the 
same ; and hence it is inferred, that the boundary which limits 
the view on all sides, is a circle. Moreover, at whatever elevation 
the dip of the horizon is taken, in any part of the earth, the 
space seen by the spectator is always circular. Hence the sur- 
face of the earth is spherical. 

10. The earth being a sphere, the dip of the horizon HOD= 
OCD. Therefore, to find the dip of the horizon corresponding 
to any given height AO* (the diameter of the earth being known,) 
we have in the triangle OCD, the right angle at D, and the two 
sides CD, CO, to find the angle OCD. Therefore, 

CO : rad. : : CD : cos. OCD. Learning the dip corresponding 
to different altitudes, by giving to the line AO different values, 
we may arrange the results in a table. 

* The learner will remark that the line AO, as drawn in the figure, is much larger 
m proportion to CA than is actually the case, and that the angle HOD is much too 
great for the reality. Such disproportions are very frequent in astronomical diagrams, 
especially when some of the parts are exceedingly small compared with others ; and 
hence the diagrams employed in astronomy are not to be regarded as true pictures of 
the magnitudes concerned, but merely as representing their abstract geometrical re. 
lotions. 



6 



THE EARTH. 



Table showing the Dip of the Horizon at different elevations, from 
1 foot to 100 feet.* 



Feet. 


/ // 


Feet. 


/ // 


Feet. 


/ // 


1 


0.59 


13 


3.33 


26 


5.01 


2 


1.24 


14 


3.41 


28 


5.13 


3 


1.42 


15 


3.49 


30 


5.23 


4 


1.58 


16 


3.56 


35 


5.49 


5 


2.12 


17 


4.03 


40 


6.14 


6 


2.25 


18 


4.11 


45 


6.36 


7 


2.36 


19 


4.17 


50 


6.58 


8 


2.47 


20 


4.24 


60 


7.37 


9 


2.57 


21 


4.31 


70 


8.14 


10 


3.07 


22 


4.37 


80 


8.48 


11 


3.16 


23 


4.43 


90 


9.20 


12 


3.25 


24 


4.49 


100 


9.51 



Such a table is of use in estimating the altitude of a body 
above the horizon, when the instrument (as usually happens) is 
more or less elevated above the general level of the earth. For 
if it is a star whose altitude above the horizon is required, the 
instrument being situated at O, (Fig. 1,) the inquiry is how far 
the star is elevated above the level HOR, but the angle taken is 
that above the visible horizon OD. The dip, therefore, or the 
angle HOD, corresponding to the height of the point O, must be 
subtracted, to obtain the true altitude. On the Peak of Tene- 
rifFe, a mountain 13,000 feet high, Humboldt observed the surface 
of the sea to be depressed on all sides nearly 2 degrees. The 
sun arose to him 12 minutes sooner than to an inhabitant of the 
plain ; and from the plain, the top of the mountain appeared en- 
lightened 12 minutes before the rising or after the setting of 
the sun. 



11. The foregoing considerations show that the form of the 
earth is spherical ; but more exact determinations prove, that the 
earth, though nearly globular, is not exactly so : its diameter from 
the north to the south pole is about 26 miles less than through 
the equator, giving to the earth the form of an oblate spheroid, f 

* This table includes the allowance for refraction. 

t An oblate spheroid is the solid described by the revolution of an ellipse about its 
shorter axis. 



FIGURE AND DIMENSIONS. 



or a flattened sphere resembling an orange. We shall reserve the 
explanations of the methods by which this fact is established, 
until the learner is better prepared than at present to understand 
them. 



12. The mean or average diameter of the earth, is 7912.4 miles, 
a measure which the learner should fix in his memory as a stand- 
ard of comparison in astronomy, and of which he should endeavor 
to form the most adequate conception in his power. The circum- 
ference of the earth is about 25,000 miles (24857.5).* Although 
the surface of the earth is uneven, sometimes rising in high moun- 
tains, and sometimes descending in deep valleys, yet these eleva- 
tions and depressions are so small in comparison with the immense 
volume of the globe, as hardly to occasion any sensible deviation 
from a surface uniformly curvilinear. The irregularities of the 
earth's surface in this view, are no greater than the rough points 
on the rind of an orange, which do not perceptibly interrupt its 
continuity ; for the highest mountain on the globe is only about 
five miles above the general level ; and the deepest mine hitherto 

5 1 

opened is only about half a mile.t Now =- — — » or about 

7912 1582 

one sixteen hundredth part of the whole diameter, an inequality 

which, in an artificial globe of eighteen inches diameter, amounts 

to only the eighty-eighth part of an inch. 



13. The diameter of the earth, con- 
sidered as a perfect sphere, may be de- 
termined by means of observations on 
a mountain of known elevation, seen 
in the horizon from the sea. Let BD 
(Fig. 2,) be a mountain of known 
height «, whose top is seen in the hori- 
zon by a spectator at A, b miles from it. 
Let x denote the radius of the earth. 
Then x~ + 6 2 = (.r+«) 2 = * 2 + 2«x + a*. 



* It will generally be sufficient to treasure up in the memory the round number, 
but sometimes, in astronomical calculations, the more exact number may be required, 
and it is therefore inserted. 

t Sir John Herschel. 




8 THE EARTH. 

I 

b 2 — c? 
Hence, 2ax=h 2 —a?, and x——- — . For example, suppose the 

height of the mountain is just one mile ; then it will be found, 

by observation, to be visible on the horizon at the distance of 

on -i l xj &2 -« 2 (89) 2 -l 7921-1 nnGn ,. 

89 miles=6. Hence, =-i — '- = =3960=radius 

2a 2 2 

of the earth, and 7920 =the earth's diameter. 



14. Another method, and the most ancient, is to ascertain the 
distance on the surface of the earth, corresponding to a degree of 
latitude. Let us select two convenient places, one lying directly 
north of the other, whose difference of latitude is known. Sup- 
pose this difference to be 1° 30', and the distance between the 
two places, as measured by a chain, to be 104 miles. Then, 
since there are 360 degrees of latitude in the entire circumference, 

24960 

1° 30' : 104 : : 360° : 24960. And - =7944. 

3.1416 

The foregoing approximations are sufficient to show that the 

earth is about 8,000 miles in diameter. 

15. The greatest difficulty in the way of acquiring correct 
views in astronomy, arises from the erroneous notions that pre- 
occupy the mind. To divest himself of these, the learner should 
conceive of the earth as a huge globe occupying a small portion 

Fig. 3. 




DOCTRINE OF THE SPHERE. 9 

of space, and encircled on all sides with the starry sphere. He 
should free his mind from its habitual proneness to consider one 
part of space as naturally up and another down, and view him- 
self as subject to a force which binds him to the earth as truly as 
though he were fastened to it by some invisible cords or wires, as 
the needle attaches itself to all sides of a spherical loadstone. He 
should dwell on this point until it appears to him as truly up in 
the direction of BB', CC, DD', (Fig. 3,) when he is at B, C, and 
D, respectively, as in the direction of AA' when he is at A. 

DOCTRINE OF THE SPHERE. 

16. The definitions of the different lines, points, and circles, 
which are used in astronomy, and the propositions founded upon 
them, compose the Doctrine of the Sphere* 

17. A section of a sphere by a plane cutting it in any manner, 
is a circle. Great circles are those which pass through the center 
of the sphere, and divide it into two equal hemispheres : Small 
circles, are such as do not pass through the center, but divide the 
sphere into two unequal parts. Every circle, whether great or 
small, is divided into 360 equal parts called degrees. A degree, 
therefore, is not any fixed or definite quantity, but only a certain 
aliquot part of any circle. 

18. The Axis of a circle, is a straight line passing through its 
center at right angles to its plane. 

19. The Pole of a great circle, is the point on the sphere where 
its axis cuts through the sphere. Every great circle has two 
poles, each of which is every where 90° from the great circle. 
For, the measure of an angle at the center of a sphere, is the 
arc of a great circle intercepted between the two lines that con- 
tain the angle ; and, since the angle made by the axis and any 
radius of the circle is a right angle, consequently its measure on 
the sphere, namely, the distance from the pole to the circumfer- 



* It is presumed that many of those who read this work, will have studied Spherical 
Geometry ; but it is so important to the student of astronomy to have a clear idea of 
the circles of the sphere, that it is thought best to introduce them here. 

2 



10 THE EARTH. 

ence of the circle, must be 90°. If two great circles cut each 
other at right angles, the poles of each circle lie in the circum- 
ference of the other circle. For each circle passes through the 
axis of the other. 

20. All great circles of the sphere cut each other in two points 
diametrically opposite, and consequently, their points of section 
are 180° apart. For the line of common section, is a diameter 
in both circles, and therefore bisects both. 

21. A great circle which passes through the pole of another 
great circle, cuts the latter at right angles. For, since it passes 
through the pole and the center of the circle, it must pass through 
the axis ; which being at right angles to the plane of the circle, 
every plane which passes through it is at right angles to the same 
plane. 

The great circle which passes through the pole of another great 
circle and is at right angles to it, is called a secondary to that circle. 

22. The angle made by two great circles on the surface of the 
sphere, is measured by the arc of another great circle, of which 
the angular point is the pole, being the arc of that great circle 
intercepted between those two circles. For this arc is the meas- 
ure of the angle formed at the center of the sphere by two radii, 
drawn at right angles to the line of common section of the two 
circles, one in one plane and the other in the other, which angle 
is therefore that of the inclination of those planes. 

23. In order to fix the position of any plane, either on the sur- 
face of the earth or in the heavens, both the earth and the heav- 
ens are conceived to be divided into separate portions by circles, 
which are imagined to cut through them i,n various ways. The 
earth thus intersected is called the terrestrial, and the heavens the 
celestial sphere. The learner will remark, that these circles have 
no existence in nature, but are mere landmarks, artificially con- 
trived for convenience of reference. On account of the immense 
distance of the heavenly bodies, they appear to us, wherever we 
are placed, to be fixed in the same concave surface, or celestial 



DOCTRINE OP THE SPHERE. 11 

vault. The great circles of the globe, extended every way to 
meet the concave surface of the heavens, become circles of the 
celestial sphere. 

24. The Horizon is the great circle which divides the earth 
into upper and lower hemispheres, and separates the visible heav- 
ens from the invisible. This is the rational horizon. The sen- 
sible horizon, is a circle touching the earth at the place of the 
spectator, and is bounded by the line in which the earth and skies 
seem to meet. The sensible horizon is parallel to the rational, 
but is distant from it by the semi-diameter of the earth, or nearly 
4,000 miles. Still, so vast is the distance of the starry sphere, 
that both these planes appear to cut that sphere in the same line ; 
so that we see the same hemisphere of stars that we should see if 
the upper half of the earth were removed, and we stood on the 
rational horizon. 

25. The poles of the horizon are the zenith and nadir. The 
Zenith is the point directly over our head, and the Nadir that di- 
rectly under our feet. The plumb line is in the axis of the hori- 
zon, and consequently directed towards its poles. 

Every place on the surface of the earth has its own horizon ; 
and the traveller has a new horizon at every step, always extend- 
ing 90 degrees from his zenith in all directions. 

26. Vertical circles are those which pass through the poles of 
the horizon, perpendicular to it. 

The Meridian is that vertical circle which* passes through the 
north and south points. 

The Prime Vertical, is that vertical circle which passes through 
the east and west points. 

27. As in geometry, we determine the position of any point by 
means of rectangular coordinates, or perpendiculars drawn from 
the point to planes at right angles to each other, so in astron- 
omy we ascertain the place of a body, as a fixed star, by taking 
its angular distance from two great circles, one of which is per- 
pendicular to the other. Thus the horizon and the meridian, or the 



12 THE EARTH. 

horizon and the prime vertical, are coordinate circles used for such 
measurements. 

The Altitude of a body, is its elevation above the horizon meas- 
ured on a vertical circle. 

The Azimuth of a body, is its distance measured on the hori- 
zon from the meridian to a vertical circle passing through the body. 

The Amplitude of a body, is its distance on the horizon, from 
the prime vertical, to a vertical circle passing through the body. 

Azimuth is reckoned 90° from either the north or south point ; 
and amplitude 90° from either the east or west point. Azimuth 
and amplitude are mutually complements of each other. When a 
point is on the horizon, it is only necessary to count the number 
of degrees of the horizon between that point and the meridian, 
in order to find its azimuth ; but if the point is above the horizon, 
then its azimuth is estimated by passing a vertical circle through 
it, and reckoning the azimuth from the point where this circle cuts 
the horizon. 

The Zenith Distance of a body is measured on a vertical cir- 
cle, passing through that body. It is the complement of the alti- 
tude. 

28. The Axis of the Earth is the diameter, on which the earth 
is conceived to turn in its diurnal revolution. The same line con- 
tinued until it meets the starry concave, constitutes the axis of the 
celestial sphere. 

The Poles of the Earth are the extremities of the earth's axis : 
the Poles of the Heavens, the extremities of the celestial axis. 

29. The Equator is a great circle cutting the axis of the earth 
at right angles. Hence the axis of the earth is the axis of the 
equator, and its poles are the poles of the equator. The intersec- 
tion of the plane of the equator with the surface of the earth, 
constitutes the terrestrial, and with the concave sphere of the 
heavens, the celestial equator. The latter, by way of distinction, 

is sometimes denominated the equinoctial. 

* 

30. The secondaries to the equator, that is, the great circles 
passing through the poles of the equator, are called Meridians, 






DOCTRINE OF THE SPHERE. 13 

because that secondary which passes through the zenith of any 
place is the meridian of that place, and is at right angles both to 
the equator and the horizon, passing as it does through the poles 
of both. (Art. 21.) These secondaries are also called Hour Circles, 
because the arcs of the equator intercepted between them are used 
as measures of time. 

31. The Latitude of a place on the earth, is its distance from 
the equator north or south. The Polar Distance, or angular dis- 
tance from the nearest pole, is the complement of the latitude. 

32. The Longitude of a place is its distance from some stand- 
ard meridian, either east or west, measured on the equator. The 
meridian usually taken as the standard, is that of the Observatory 
of Greenwich, near London. If a place is directly on the equator, 
we have only to inquire how many degrees of the equator there 
are between that place and the point where the meridian of Green- 
wich cuts the equator. If the place is north or south of the equa- 
tor, then its longitude is the arc of the equator intercepted between 
the meridian which passes through the place, and the meridian of 
Greenwich. 

33. The Ecliptic is a great circle in which the earth performs 
its annual revolution around the sun. It passes through the center 
of the earth and the center of the sun. It is found by observa- 
tion that the earth does not lie with its axis at right angles to the 
plane of the ecliptic, but that it is turned about 23| degrees out of 
a perpendicular direction, making an angle with the plane itself of 
66£°. The equator, therefore, must be turned the same distance 
out of a coincidence with the ecliptic, the two circles making 
an angle with* each other of 23^°, (23° 27' 40".) It is particu- 
larly important for the learner to form correct ideas of the eclip- 
tic, and of its relations to the equator, since to these two circles a 
great number of astronomical measurements and phenomena are 
referred. 

34. The Equinoctial Points, or Equinoxes,* are the intersec- 

* The term Equinoxes strictly denotes the times when the sun arrives at the equi- 
noctial points, but it is also frequently used to denote those points themselves. 



14 THE EARTH. 

tions of the ecliptic and equator. The time when the sun crosses 
the equator in returning northward is called the vernal, and in 
going southward, the autumnal equinox. The vernal equinox 
occurs about the 21st of March, and the autumnal the 22d of 
September. 

35. The Solstitial Points are the two points of the ecliptic 
most distant from the equator. The times when the sun comes 
to them are called solstices. The summer solstice occurs about 
the 22d of June, and the winter solstice about the 22d of De- 
cember. 

The ecliptic is divided into twelve equal parts of 30° each, 
called signs, which, beginning at the vernal equinox, succeed each 
other in the following order : 



Northern. 


Southern. 


1. Aries T 


7. Libra =£= 


2. Taurus b 


8. Scorpio fli 


3. Gemini II 


9. Sagittarius / 


4. Cancer 2o 


10. Capricornus V3 


5. Leo SI 


11. Aquarius ~ 


6. Virgo tt# 


12. Pisces x 



The mode of reckoning on the ecliptic, is by signs, degrees, 
minutes, and seconds. The sign is denoted either by its name 
or its number. Thus 100° may be expressed either as the 10th 
degree of Cancer, or as 3 s 10°. 

36. Of the various meridians, two are distinguished by the 
name of Colures. The Equinoctial Colure, is the meridian which 
passes through the equinoctial points. The Solstitial Colure, is 
the meridian which passes through the solstitial points. As the 
solstitial points are 90° from the equinoctial points, so the sol- 
stitial colure is 90° from the equinoctial colure. It is also at right 
angles, or a secondary to both the ecliptic and equator. For like 
every other meridian, it is of course perpendicular to the equator, 
passing through its poles. Moreover, the equinox, being a point 
both in the equator and in the ecliptic, is 90° from the solstice, 
from the pole of the equator, and from the pole of the ecliptic. 



DOCTRINE OF THE SPHERE. 



15 



Hence the solstitial colure, which passes through the solstice and 
the pole of the equator, passes also through the pole of the ecliptic, 
being the great circle of which the equinox itself is the pole. 
Consequently, the solstitial colure is a secondary to both the equa- 
tor and the ecliptic. (See Arts. 19, 20, 21.) 

37. The position of a celestial body is referred to the equator 
by its right ascension and declination. (See Art. 27.) Right 
Ascension, is the angular distance from the vernal equinox, meas- 
ured on the equator. If a star is situated on the equator, then its 
right ascension is the number of degrees of the equator between 
the star and the vernal equinox. But if the star is north or south 
of the equator, then its right ascension is the arc of the equator 
intercepted between the vernal equinox and that secondary to the 
equator which passes through the star. Declination is the dis- 
tance of a body from the equator, measured on a secondary to the 
latter. Therefore, right ascension and declination correspond to 
terrestrial longitude and latitude, right ascension being reckoned 
from the equinoctial colure, in the same manner as longitude is 
reckoned from the meridian of Greenwich. On the other hand, 
celestial longitude and latitude are referred, not to the equator, 
but to the ecliptic. Celestial Longitude, is the distance of a body 
from the vernal equinox reckoned on the ecliptic. Celestial Lati- 
tude, is distance from the ecliptic measured on a secondary to the 
latter. Or, more briefly, Longitude is distance on the ecliptic ; 
Latitude, distance from the ecliptic. The North Polar Distance 
of a star, is the complement of its declination. 



38. Parallels of Latitude are small 
circles parallel to the equator. They ' 
constantly diminish in size as we go 
from the equator to the pole, the ra- 
dius being always equal to the cosine 
of the latitude. In fig. 4, let HO be 
the horizon, EQ the equator, PP the 
axis of the earth, ZN the prime ver- 
tical, and ZL a parallel of latitude of 
any place Z. Then ZE is the lati- 




16 THE EARTH. 

tude, (Art. 31,) and ZP the complement of the latitude ; but Zn 
the radius of the parallel of latitude ZL, is the sine of ZP, and 
therefore the cosine of the latitude. 

39. The Tropics are the parallels of latitude that pass through 
the solstices. The northern tropic is called the tropic of Cancer ; 
the southern, the tropic of Capricorn. 

40. The Polar Circles are the parallels of latitude that pass 
through the poles of the ecliptic, at the distance of 23| degrees 
from the pole of the earth. (Art. 33.) 

41. The earth is divided into five zones. That portion of the 
earth which lies between the tropics, is called the Torrid Zone ; 
that between the tropics and polar circles, the Temperate Zones ; 
and that between the polar circles and the poles, the Frigid 
Zones. 

42. The Zodiac is the part of the celestial sphere which lies 
about 8 degrees on each side of the ecliptic. This portion of the 
heavens is thus marked off by itself, because the planets are never 
seen further from the ecliptic than this limit. 

43. The elevation of the pole is equal to the latitude of the place. 
The arc PE (Fig. 4.)=ZO.\PO=ZE which equals the lati- 
tude. 

44. The elevation of the equator is equal to the complement of 
the latitude. 

ZH=90°. But ZE=Lat..\EH=90— Lat.=colatitude. 

45. The distance of any place from the pole (or the polar dis- 
tance) equals the complement of the latitude. 

EP=90°. ButEZ=Lat..\ZP=90-Lat.=colatitude. 



DIURNAL REVOLUTION. 17 



CHAPTER II. 

DIURNAL REVOLUTION ARTIFICIAL GLOBES— ASTRONOMICAL 

PROBLEMS. 

46. The apparent diurnal revolution of the heavenly bodies 
from east to west, is owing to the actual revolution of the earth 
on its own axis from west to east. If we conceive of a radius of 
the earth's equator extended until it meets the concave sphere of 
the heavens, then as the earth revolves, the extremity of this line 
would trace out a curve on the face of the sky, namely, the celes- 
tial equator. In curves parallel to this, called the circles of diurnal 
revolution, the heavenly bodies actually appear to move, every star 
having its own peculiar circle. After the learner has first rendered 
familiar the real motions of the earth from west to east, he may then, 
without danger of misconception, adopt the common language, 
that all the heavenly bodies revolve around the earth once a day 
from east to west, in circles parallel to the equator and to each other. 

47. The time occupied by a star in passing from any point in 
the meridian until it comes round to the same point again, is called 
a sidereal day, and measures the period of the earth's revolution 
on its axis. If we watch the returns of the same star from day to 
day, we shall find the intervals exactly equal to one another ; 
that is, the sidereal days are all equal* Whatever star we select 
for the observation, the same result will be obtained. The stars, 
therefore, always keep the same relative position, and have a 
common movement round the earth, — a consequence that natu- 
rally flows from the hypothesis, that their apparent motion is all 
produced by a single real motion, namely, that of the earth. The 
sun, moon, and planets, revolve in like manner, but their returns to 
the meridian are not, like those of the fixed stars, at exactly equal 
intervals. 

48. The appearances of the diurnal motions of the heavenly 

* Allowance is here supposed to be made for the effects of precession, &c. 

3 



18 THE EARTH. 

bodies are different in different parts of the earth, since every 
place has its own horizon, (Art. 15,) and different horizons are 
variously inclined to each other. Let us suppose the spectator 
viewing the diurnal revolutions, successively, from several different 
positions on the earth. 

49. If he is on the equator, his horizon passes through both poles ; 
for the horizon cuts the celestial vault at 90 degrees in every di- 
rection from the zenith of the spectator ; but the pole is likewise 
90 degrees from his zenith, and consequently, the pole must be 
in his horizon. The celestial equator coincides with his Prime 
Vertical, being a great circle passing through the east and 
west points. Since all the diurnal circles are parallel to the equa- 
tor, they are all, like the equator, perpendicular to his horizon. 
Such a view of the heavenly bodies, is called a right sphere ; or, 

A Right Sphere is one in which all the daily revolutions of 
the heavenly bodies are in circles perpendicular to the horizon. 

A right sphere is seen only at the equator. Any star situated 
in the celestial equator, would appear to rise directly in the east, at 
noon to pass through the zenith of the spectator, and to set directly in 
the west ; in proportion as stars are at a greater distance from the 
equator towards the pole, they describe smaller and smaller circles, 
until, near the pole, their motion is hardly perceptible. In a right 
sphere every star remains an equal time above and below the hori- 
zon ; and since the times of their revolutions are equal, the veloci- 
ties are as the lengths of the circles they describe. Consequently, 
as the stars are more remote from the equator towards the pole, 
their motions become slower, until, at the pole, the north star ap- 
pears stationary. 

50. If the spectator advances one degree towards the north 
pole, his horizon reaches one degree beyond the pole of the earth, 
and cuts the starry sphere one degree below the pole of the heav- 
ens, or below the north star, if that be taken as the place of the 
pole. As he moves onward towards the pole, his horizon contin- 
ually reaches further and further beyond it, until when he comes 
to the pole of the earth, and under the pole of the heavens, his 
horizon reaches on all sides to the equator and coincides with it. 



DIURNAL REVOLUTION. 19 

Moreover, since all the circles of daily motion are parallel to the 
equator, they become, to the spectator at the pole, parallel to the 
horizon. This is what constitutes a parallel sphere. Or, 

A Parallel Sphere is that in which all the circles of daily 
motion are parallel to the horizon. 

51. To render this view of the heavens familiar, the learner 
should follow round in his mind a number of separate stars, one 
near the horizon, one a few degrees above it, and a third near the 
zenith. To one who stood upon the north pole, the stars of the 
northern hemisphere would all be perpetually in view when not 
obscured by clouds or lost in the sun's light, and none of those of 
the southern hemisphere would ever be seen. The sun would 
be constantly above the horizon for six months in the year, and 
the remaining six constantly out of sight. That is, at the pole 
the days and nights are each six months long. The phenomena 
at the south pole are similar to those at the north. 

52. A perfect parallel sphere can never be seen except at one 
of the poles, — a point which has never been actually reached by 
man; yet the British discovery ships penetrated within a few 
degrees of the north pole, and of course enjoyed the view of a 
sphere nearly parallel. 

53. As the circles of daily motion are parallel to the horizon of 
the pole, and perpendicular to that of the equator, so at all places 
between the two, the diurnal motions are oblique to the horizon. 
This aspect of the heavens constitutes an oblique sphere, which is 
thus defined : 

An Oblique Sphere is that in which the circles of daily mo- 
tion are oblique to the horizon. 

Suppose for example the spectator is at the latitude of fifty de- 
grees. His horizon reaches 50° beyond the pole of the earth, and 
gives the same apparent elevation to the pole of the heavens. It 
cuts the equator, and all the circles of daily motion, at an angle 
of 40°, being always equal to the co-altitude of the pole. Thus, 
let HO (Fig. 5,) represent the horizon, EQ the equator, and 
PP' the axis of the earth. Also, //, mm, &c. parallels of latitude. 



20 



THE EARTH. 




Then the horizon of a spectator Fi g- 5 - 

at Z, in latitude 50° reaches to 
50° beyond the pole (Art. 50) ; 
and the angle ECH, is 40°. As 
we advance still further north, 
the elevation of the diurnal cir- 
cles grows less and less, and 
consequently .the motions of the 
heavenly bodies more and more 
oblique, until finally, at the pole, 
where the latitude is 90°, the 
angle of elevation of the equator 
vanishes, and the horizon and equator coincide with each other, 
as before stated. 

54. The circle of perpetual apparition, is the boundary of 
that space around the elevated pole, where the stars never set. 
Its distance from the pole is equal to the latitude of the place. 
For, since the altitude of the pole is equal to the latitude, a star 
whose polar distance is just equal to the latitude, will when at its 
lowest point only just reach the horizon ; and all the stars nearer 
the pole than this will evidently not descend so far as the horizon. 

Thus, mm (Fig. 5.) is the circle of perpetual apparition, be- 
tween which and the north pole, the stars never set, and its dis- 
tance from the pole OP is evidently equal to the elevation of the 
pole, and of course to the latitude. 

55. In the opposite hemisphere, a similar part of the sphere 
adjacent to the depressed pole never rises. Hence, 

The circle of perpetual occultation, is the boundary of that 
space around the depressed pole, within which the stars never rise. 
Thus, m'm' (Fig. 5,) is the circle of perpetual occultation, be- 
tween which and the south pole, the stars never rise. 



56. In an oblique sphere, the horizon cuts the circles of daily 
motion unequally. Towards the elevated pole, more than half 
the circle is above the horizon, and a greater and greater portion 
as the distance from the equator is increased, until finally, within 



DIURNAL REVOLUTION. 21 

the circle of perpetual apparition, the whole circle is above the 
horizon. Just the opposite takes place in the hemisphere next 
the depressed pole. Accordingly, when the sun is in the equator, 
as the equator and horizon, like all other great circles of the 
sphere, bisect each other, the days and nights are equal all over 
the globe. But when the sun is north of the equator, our days 
become longer than our nights, but shorter when the sun is 
south of the equator. Moreover, the higher the latitude, the 
greater is the inequality in the lengths of the days and nights. 
All these points will be readily understood by inspecting figure 5. 

57. Most of the phenomena of the diurnal revolution can be 
explained, either on the supposition that the celestial sphere actu- 
ally all turns around the earth once in 24 hours, or that this mo- 
tion of the heavens is merely apparent, arising from the revolu- 
tion of the earth on its axis in the opposite direction, — a motion 
of which we are insensible, as we sometimes lose the conscious- 
ness of our own motion in a ship or a steamboat, and observe all 
external objects to be receding from us with a common motion. 
Proofs entirely conclusive and satisfactory, establish the fact, that 
it is the earth and not the celestial sphere that turns ; but these 
proofs are drawn from various sources, and the student is not pre- 
pared to appreciate their value, or even to understand some of 
them, until he has made considerable proficiency in the study of 
astronomy, and become familiar with a great variety of astronom- 
ical phenomena. To such a period of our course of instruction, 
we therefore postpone the discussion of the hypothesis .of the 
earth's rotation on its axis. 

58. While we retain the same place on the earth, the diurnal 
revolution occasions no change in our horizon, but our horizon 
goes round as well as ourselves. Let us first take our station on 
the equator at sunrise ; our horizon now passes through both the 
poles, and through the sun, which we are to conceive of as at a 
great distance from the earth, and therefore as cut, not by the 
terrestrial but by the celestial horizon. As the earth turns, the 
horizon dips more and more below the sun, at the rate of 15 de- 
grees for every hour, and, as in the case of the polar star, (Art. 50,) 



22 THE EARTH. 

the sun appears to rise at the same rate. In six hours, therefore, 
it is depressed 90 degrees below the sun, which brings us directly 
under the sun, which, for our present purpose, we may consider as 
having all the while maintained the same fixed position in space. 
The earth continues to turn, and in six hours more, it completely 
reverses the position of our horizon, so that the western part of 
the horizon which at sunrise was diametrically opposite to the 
sun now cuts the sun, and soon afterwards it rises above the level 
of the sun, and the sun sets. During the next twelve hours, the 
sun continues on the invisible side of the sphere, until the hori- 
zon returns to the position from which it started, and a new day 
begins. 

59. Let us next contemplate the similar phenomena at the poles. 
Here the horizon, coinciding as it does with the equator, would 
cut the sun through its center, and the sun would appear to re- 
volve along the surface of the sea, one half above and the other 
half below the horizon. This supposes the sun in its annual 
revolution to be at one of the equinoxes. When the sun is north 
of the equator, it revolves continually round in a path which, 
during a single revolution, appears parallel to the equator, and it 
is constantly day ; and when the sun is south of the equator, it is, 
for the same reason, continual night. 

60. We have endeavored to conceive of the manner in which 
the apparent diurnal movements of the sun are really produced at 
two stations, namely, in the right sphere, and in the parallel sphere. 
These two cases being clearly understood, there will be little dif- 
ficulty in applying a similar explanation to an oblique sphere. 

ARTIFICIAL GLOBES. 

61. Artificial globes are of two kinds, terrestrial and celestial. 
The first exhibits a miniature representation of the earth ; the 
second, of the visible heavens ; and both show the various circles 
by which the two spheres are respectively traversed. Since all 
globes are similar solid figures, a small globe, imagined to be sit- 
uated at the center of the earth or of the celestial vault, may rep- 



ARTIFICIAL GLOBES. 23 

resent all the visible objects and artificial divisions of either sphere, 
and with great accuracy and just proportions, though on a scale 
greatly reduced. The study of artificial globes, therefore, cannot 
be too strongly recommended to the student of astronomy.* 

62. An artificial globe is encompassed from north to south by 
a strong brass ring to represent the meridian of the place. This 
ring is made fast to the two poles and thus supports the globe, 
while it is itself supported in a vertical position by means of a 
frame, the ring being usually let into a socket in which it may be 
easily slid, so as to give any required elevation to the pole. The 
brass meridian is graduated each way from the equator to the 
pole 90°, to measure degrees of latitude or declination, according 
as the distance from the equator refers to a point on the earth or 
in the heavens. The horizon is represented by a broad zone, made 
broad for the convenience of carrying on it a circle of azimuth, an- 
other of amplitude, and a wide space on which are delineated the 
signs of the ecliptic, and the sun's place for every day in the year ; 
not because these points have any special connexion with the hori- 
zon, but because this broad surface furnishes a convenient place 
for recording them. 

63. Hour Circles are represented on the terrestrial globe by 
great circles drawn through the pole of the equator ; but, on the 
celestial globe, corresponding circles pass through the poles of the 
ecliptic, constituting circles of celestial latitude, (Art. 37,) while the 
brass meridian, being a secondary to the equinoctial, becomes an 
hour circle of any star which, by turning the globe, is brought un- 
der it. 

64. The Hour Index is a small circle described around the pole 
of the equator, on which are marked the hours of the day. As 
this circle turns along with the globe, it makes a complete revo- 
lution in the same time with the equator ; or, for any less period, 

* It were desirable, indeed, that every student of the science should have the celes- 
tial globe at least, constantly before him. One of a small size, as eight or nine inches, 
will answer the purpose, although globes of these dimensions cannot usually be relied 
on for nice measurements. 



24 THE EARTH. 

the same number of degrees of this circle and of the equator pass 
under the meridian. Hence the hour index measures arcs of 
right ascension. (Art. 37.) 

65. The Quadrant of Altitude is a flexible strip of brass, gradu- 
ated into ninety equal parts, corresponding in length to degrees 
on the globe, so that when applied to the globe and bent so as 
closely to fit its surface, it measures the angular distance between 
any two points. When the zero, or the point where the gradua- 
tion begins, is laid on the pole of any great circle, the 90th degree 
will reach to the circumference of that circle, and being therefore 
a great circle passing through the pole of another great circle, it 
becomes a secondary to the latter. (Art. 21 .) Thus the quadrant 
of altitude may be used as a secondary to any great circle on the 
sphere ; but it is used chiefly as a secondary to the horizon, the 
point marked 90° being screwed fast to the pole of the horizon, 
that is, the zenith, and the other end, marked 0, being slid along 
between the surface of the sphere and the wooden horizon. It 
thus becomes a vertical circle, on which to measure the altitude 
of any star through which it passes, or from which to measure 
the azimuth of the star, which is the arc of the horizon intercept- 
ed between the meridian and the quadrant of altitude passing 
through the star, (Art. 27.) 

66. To rectify the globe for any place, the north pole must be 
elevated to the latitude of the place (Art. 43) ; then the equator 
and all the diurnal circles will have their due inclination in respect 
to the horizon ; and, on turning the globe, (the celestial globe west, 
and the terrestrial east,) every point on either globe will revolve as 
the same point does in nature ; and the relative situations of all 
places will be the same as on the respective native spheres. 

PROBLEMS ON THE TERRESTRIAL GLOBE. 

67. To find the Latitude and Longitude of a place : Turn the 
globe so as to bring the place to the brass meridian ; then the de- 
gree and minute on the meridian directly over the place will indi- 
cate its latitude, and the point of the equator under the meridian, 
will show its longitude. 



PROBLEMS ON THE TERRESTRIAL GLOBE. 25 

Ex. What are the Latitude ^and Longitude of the city of New 
York? 

68. To find a place having its latitude and longitude given : Bring 
to the brass meridian the point of the equator corresponding to 
the longitude, and then at the degree of the meridian denoting the 
latitude, the place will be found. 

Ex. What place on the globe is in Latitude 39 N. and Longi- 
tude 77 W. ? 

69. To find the bearing and distance of two places: Rectify the 
globe for one of the places (Art. 66) ; screw the quadrant of alti- 
tude to the zenith,* and let it pass through the other place. Then 
the azimuth will give the bearing of the second place from the 
first, and the number of degrees on the quadrant of altitude, mul- 
tiplied by 69^, (the number of miles in a degree,) will give the 
distance between the two places. 

Ex. What is the bearing of New Orleans from New York, and 
what is the distance between these places ? 

70. To determine the difference of time in different places : 
Bring the place that lies eastward of the other to the meridian, 
and set the hour index at XII. Turn the globe eastward until 
the other place comes to the meridian, then the index will point 
to the hour required. 

Ex. When it is noon at New York, what time is it at London ? 

71. The hour being given at any place, to tell what hour it is in 
any other part of the world : Find the difference of time between 
the two places, (Art. 70,) and, if the place whose time is required 
is eastward of the other, add this difference to the given time, but 
if westward, subtract it. 

Ex. What time is it at Canton, in China, when it is 9 o'clock 
A. M. at New York ? 

72. To find the antozci,\ the pericecal and the antipodes^ of any 

* The zenith will of course be the point of the meridian over the place, 
t avn otKos- \ rrepi oiko{. § avn jrtrj. 

4 



26 THE EARTH. 

place : Bring the given place to the meridian ; then, under the 
meridian, in the opposite hemisphere, in the same degree of lati- 
tude, will be found the antceci. The same place remaining under 
the meridian, set the index to XII, and turn the globe until the 
other XII is under the index ; then the perioeci will be on the me- 
ridian, under the same degree of latitude with the given place, 
and the antipodes will be under the meridian, in the same latitude, 
in the opposite hemisphere. 

Ex. Find the antoeci, the perioeci, and the antipodes of the citi- 
zens of New York. 

The antoeci have the same hour of the day, but different seasons 
of the year ; the perioeci have the same seasons, but opposite hours ; 
and the antipodes have both opposite hours and opposite seasons. 

73. To rectify the globe for the surfs place: On the wooden 
horizon, find the day of the month, and against it is given the sun's 
place in the ecliptic, expressed by signs and degrees.* Look for 
the same sign and degree on the ecliptic, bring that point to the 
meridian and set the hour index to XII. To all places under the 
meridian it will then be noon. 

Ex. Rectify the globe for the sun's place on the 1st of September. 

74. The latitude of the place being given, to find the time of the 
surfs rising and setting on any given day at that place : Having 
rectified the globe for the latitude, (Art. 66,) bring the sun's place 
in the ecliptic to the graduated edge of the meridian, and set the 
hour index to XII ; then turn the globe so as to bring the sun to 
the eastern and then to the western horizon, and the hour index 
will show the times of rising and setting respectively. 

Ex. At what time will the sun rise and set at New Haven, 
Lat. 41° 18', on the 10th of July? 

PROBLEMS ON THE CELESTIAL GLOBE. 

75. To find the Declination and Right Ascension of a heavenly 
body : Bring the place of the body (whether the sun or a star) to 
the meridian. Then the degree and minute standing over it will 

* The larger globes have the day of the month marked against the corresponding 
Bign on the ecliptic itself. 



PROBLEMS ON THE CELESTIAL GLOBE. 27 

show its declination, and the point of the equinoctial under the 
meridian will give its right ascension. It will be remarked, that 
the declination and right ascension are found in the same manner 
as latitude and longitude on the terrestrial globe. Right ascen- 
sion is expressed either in degrees or in hours ; both being reck- 
oned from the vernal equinox, (Art. 37.) 

Ex. What is the declination and right ascension of the bright 
star Lyra ? — also of the sun on the 5th of June ? 

76. To represent the appearance of the heavens at any time: 
Rectify the globe for the latitude, bring the suns place in the 
ecliptic to the meridian, and set the hour index to XII ; then turn 
the globe westward until the index points to the given hour, and 
the constellations would then have the same appearance to an eye 
situated at the center of the globe, as they have at that moment 
in the sky. 

Ex. Required the aspect of the stars at New Haven, Lat. 41° 
18', at 10 o'clock, on the evening of December 5th. 

77. To find the altitude and azimuth of any star : Rectify the 
globe for the latitude, and let the quadrant of altitude be screwed 
to the zenith, and be made to pass through the star. The arc on 
the quadrant, from the horizon to the star, will denote its altitude, 
and the arc of the horizon from the meridian to the quadrant, will 
be its azimuth. 

Ex. What are the altitude and azimuth of Sirius (the brightest 
of the fixed stars) on the 25th of December at 10 o'clock in the 
evening, in Lat. 41° ? 

78. To find the angular distance of two stars from each other : 
Apply the zero mark of the quadrant of altitude to one of the 
stars, and the point of the quadrant which falls on the other star, 
will show the angular distance between the two. 

Ex. What is the distance between the two largest stars of the 
Great Bear ?* 

* These two stars are sometimes called " the Pointers," from the line which passes 
through them being always nearly in the direction of the north star. The angular 
distance between them is about 5°, and may be learned as a standard for reference in 
estimating, by the eye, the distance between any two points on the celestial vault. 



28 



PARALLAX. 



79. To find the surfs meridian altitude, the latitude and day 
of the month being given : Having rectified the globe for the 
latitude, (Art. 66,) bring the sun's place in the ecliptic to the me- 
ridian, and count the number of degrees and minutes between 
that point of the meridian and the zenith. The complement of 
this arc will be the sun's meridian altitude. 

Ex. What is the sun's meridian altitude at noon on the 1st of 
August, in Lat. 41° 18'? 



CHAPTER III. 



OF PARALLAX, REFRACTION, AND TWILIGHT. 



80. Parallax is the apparent change of place which bodies 
undergo by being viewed from different points. Thus in figure 
6, let A represent the earth, CH' the horizon. H'Z a quadrant of 





& 


Fig. 6. 




H 


^^gG- 


A 


K 






P^ 


-~ — _., 



a great circle of the heavens, extending from the horizon to the 
zenith ; and let E, F, G, H, be successive positions of the moon 
at different elevations, from the horizon to the meridian. Now a 
spectator on the surface of the earth at A, would refer the place 
of E to h, whereas, if seen from the center of the earth, it would 



PARALLAX. 29 

appear at H'. The arc K'h is called the parallactic arc, and the 
angle HEA, or its equal AEC, is the angle of parallax. The 
same is true of the angles at F, G, and H, respectively. 

81. Since then a heavenly body is liable to be referred to dif- 
ferent points on the celestial vault, when seen from different parts 
of the earth, and thus some confusion occasioned in the deter- 
mination of points on the celestial sphere, astronomers have agreed 
to consider the true place of a celestial object to be that where it 
would appear if seen from the center of the earth. The doctrine 
of parallax teaches how to reduce observations made at any place 
on the surface of the earth, to such as they would be if made 
from the center. 

82. The angle AEC is called the horizontal parallax, which 
may be thus defined. Horizontal Parallax, is the change of po- 
sition which a celestial body, appearing in the horizon as seen 
from the surface of the earth, would assume if viewed from the 
earth's center. It is the angle subtended by the semi-diameter 
of the earth, as viewed from the body itself. If we consider any 
one of the triangles represented in the figure, ACG for example, 

, Sin. AGC : Sin. GAZ : : AC : CG 

Q . p „ Sin. GAZ x AC Sin. GAZ 

.*. Sin. Parallax= oo 

CG CG 

Hence the sine of the angle of parallax, or (since the angle of 
parallax is always very small*) the parallax itself varies as the 
sine of the zenith distance of the body directly, and the distance 
of the body from the center of the earth inversely. Parallax, there- 
fore, increases as a body approaches the horizon, (but increasing 
with the sines, it increases much slower than in the simple ratio 
of the distance from the zenith,) and diminishes, as the distance 
from the spectator increases. Again, since the parallax AGC is as 
the sine of the zenith distance, let P represent the horizontal par- 
allax, and P' the parallax at any altitude ; then, 

* The moon, on account of its nearness to the earth, has the greatest horizontal 
parallax of any of the heavenly bodies ; yet this is less than 1° (being 57') while the 
greatest parallax of any of the planets does not exceed 30". The difference in an 
arc of 1°, between the length of the arc and the sine, is only 0."18. 



30 THE EARTH. 

P' 

F : P::sin. zenith dist. : sin. 90°.\P=— p— ' 

sin. zen. dist. 

Hence, the horizontal parallax of a body equals its parallax at 
any altitude, divided by the sine of its distance from the zenith. 

83. From observations, therefore, on the parallax of a body at 
any elevation, we are enabled to find the angle subtended by the 
semi-diameter of the earth as seen from the body. Or if the 
horizontal parallax is given, the parallax at any altitude may be 
found, for 

P'=Pxsin. zenith distance. 

Hence, in the zenith the parallax is nothing, and is at its max- 
imum in the horizon. 

84. It is evident from the figure, that the effect of parallax 
upon the place of a celestial body is to depress it. Thus, in con- 
sequence of parallax, E is depressed by the arc H'A ; F by the 
arc Vp ; G by the arc Rr ; while H sustains no change. Hence, 
in all calculations respecting the altitude of the sun, moon, or plan- 
ets, the amount of parallax is to be added ; the stars, as we shall 
see hereafter, have no sensible parallax. As the depression which 
arises from parallax is in the direction of a vertical circle, a body, 
when on the meridian, has only a parallax in declination ; but 
in other situations, there is at the same time a parallax in 
declination and right ascension ; for its direction being oblique 
to the equinoctial, it can be resolved into two parts, one of which 
(the declination) is perpendicular, and the other (the right ascen- 
sion) is parallel to the equinoctial. 

85. The mode of determining the horizontal parallax, is as 
follows : 

Let O, O', (Fig. 7,) be two places on the earth, situated under 
the same meridian, at a great distance from each other ; one place, 
for example, at the Cape of Good Hope, and the other in the north 
of Europe. The latitude of each place being known, the arc of 
the meridian 00' is known, and the angle OCO' also is known. 
Let the celestial body M, (the moon for example,) he observed 
simultaneously at O and O', and its zenith distance at each place 



PARALLAX. 



31 



accurately taken, namely, ZY and 
Z Y' ; then the angles ZOM and 
Z'O'M, and of course their sup- 
plements COM,CO'M are found. 
Then in the quadrilateral figure 
COMO', we have all the angles 
and the two radii, CO, CO', 
whence by joining 00', the side 
CM may be easily found. But, 
CM : CO : : sin. ZOM : sin. 
CMO=sine of the angle of par- 
allax ; or (since the arc is very 
small) equals the parallax P'. 
But when M as seen from O is in the horizon, ZOM becomes a 
right angle, and its sine equal to radius. Then, 

CO 

CM : CO : : 1 : P=horizontal parallax^—. 

On this principle, the horizontal parallax of the moon was de- 
termined by La Caille and La Lande, two French astronomers, 
one stationed at the Cape of Good Hope, the other at Berlin ; and 
in a similar way the parallax of Mars was ascertained, by ob- 
servations made simultaneously at the Cape of Good Hope and 
at Stockholm. 




86. On account of the great distance of the sun, his horizontal 
parallax, which is only 8". 6, cannot be accurately ascertained by 
this method. It can, however, be determined by means of the 
transits of Venus, a process to be described hereafter. 



87. The determination of the horizontal parallax of a celestial 
body is an element of great importance, since it furnishes the 
means of estimating the distance of the body from the center of 
the earth. Thus, if the angle AEC (Fig. 6,) be found, the radius 
of the earth AC being known, we have in the triangle AEC, 
right angled at A, the side AC and all the angles, to find the hypo- 
thenuse CE, which is the distance of the moon from the center 
of the earth. 



THE EARTH. 



REFRACTION. 



88. While parallax depresses the celestial bodies subject to it, 
refraction elevates them ; and it affects alike the most distant 
as well as nearer bodies, being occasioned by the change of di- 
rection which light undergoes in passing through the ' atmos- 
phere. Let us conceive of the atmosphere as made up of a great 
number of concentric strata, as A A, BB, CC, and DD, (Fig. 8,) 

Fig. 8. 




increasing rapidly in density (as is known to be the fact) in ap- 
proaching near to the surface of the earth. Let S be a star, from 
which a ray of light Sa enters the atmosphere at a, where, being 
turned towards the radius of the convex surface, it would change 
its direction into the line ab, and again into be, and cO, reach- 
ing the eye at O. Now, since an object always appears in the 
direction in which the light finally strikes the eye, the star would 
be seen in the direction of the last ray cO, and the star would 
apparently change its place, in consequence of refraction, from 
S to S', being elevated out of its true position. Moreover, 
since on account of the constant increase of density in descend- 
ing through the atmosphere, the light would be continually turned 
out of its course more and more, it would therefore move, not 
in the polygon represented in the figure, but in a corresponding 
curve, whose curvature is rapidly increased near the surface of 
the earth. 

89. When a body is in the zenith, since a ray of light from it 
enters the atmosphere at right angles to the refracting medium, it 
suffers no refraction. Consequently, the position of the heavenly 



REFRACTION. 



33 



bodies, when in the zenith, is not changed by refraction, while, 
near the horizon, where a ray of light strikes the medium very 
obliquely, and traverses the atmosphere through its densest part, 
the refraction is greatest. The following numbers, taken at dif- 
ferent altitudes, will show how rapidly refraction diminishes from 
the horizon upwards. The amount of refraction at the horizon 
is 34' 00". At different elevations it is as follows. 



Elevation. 


Refraction. 


Elevation. 


Refraction. 


0° 10' 


32' 00" 


30° 


V 40" 


20 


30 00 


40 


1 09 


1 00 


24 25 


45 


58 


5 00 


10 00 


60 


33 


10 00 


5 20 


80 


10 


20 00 


2 39 


90 


00 



From this table it appears, that while" refraction at the horizon 
is 34 minutes, at so small an elevation as only 10 minutes above 
the horizon it loses 2 minutes, more than the entire change from 
the elevation of 30° to the zenith. From the horizon to 1° above, 
the refraction is diminished nearly 10 minutes. The amount at 
the horizon, at 45°, and at 90°, respectively, is 34', 58", and 0. In 
finding the altitude of a heavenly body, the effect of parallax must 
be added, but that of refraction subtracted. 



90. Let us now learn the method, by which the amount of re- 
fraction at different elevations is ascertained. To take the sim- 
plest case, we will suppose ourselves in a high latitude, where 
some of the stars within the circle of perpetual apparition pass 
through the zenith of the place. We measure the distance of 
such a star from the pole when on the meridian above the pole, 
that is, in the zenith, where it is not at all affected by refraction, 
and again its distance from the pole in its lower culmination. 
Were it not for refraction, these two polar distances would be 
equal, since, in the diurnal revolution of a star, it is in fact always 
at the same distance from the pole ; but, on account of refraction, 
the lower distance will be less than the upper, and the difference 
between the two will show the amount of refraction at the lower 
culmination, the latitude of the place being known. 

Example. At Paris, latitude 48° 50', a star was observed to 
5 



34 



THE EARTH. 



pass the meridian 6' north of the zenith, and consequently, 41° 4', 
from the pole.* It should have passed the meridian at the same 
distance below the pole, but the distance was found to be only 
40° 57' 35". Hence, 41° 4' -40° 57' 35" =6' 25" is the refraction 
due to that altitude, that is, at the altitude of 7° 46'=(48° 50' — 
41° 4'). By taking similar observations in various places situated 
in high latitudes, the amount of refraction might be ascertained 
for a number of different altitudes, and thus the law of increase 
in refraction as we proceed from the zenith towards the horizon, 
might be ascertained. 

91. Another method of finding the refraction at different alti- 
tudes, is as follows. Take the altitude of the sun or a star, whose 
right ascension and declination are known, and note the time by 
the clock. Observe also when it crosses the meridian, and the 
difference of time between the two observations will give the hour 
angle ZP#, (Fig. 9.) In this triangle ZP# we also know PZ the 

Fig. 9. 




co-latitude and Tx the co-declination. Hence we can find the co- 
altitude Zx, and of course the true altitude. Compare the alti- 
tude thus found with that before determined by observation, and 
the difference will be the refraction due to the apparent altitude. 

* For the polar distance of the place=90-48° 50'=41° 10' ; and 41° 10'-6'= 
41° 4'. 



REFRACTION. 35 

Ex. On May 1, 1738, at 5h. 20m. in the morning, Cassini ob- 
served the altitude of the sun's center at Paris to be 5° 0' 14". The 
latitude of Paris being 48° 50' 10", and the sun's declination at 
that time being 15° 0' 25" : Required the refraction. 

By spherical trigonometry, Zx will be found equal to 85° 10' 
8" ; consequently, the true altitude was 4° 49' 52". Now to 5° 
0' 14", the apparent altitude, 9" must be added for parallax, 
and we have 5° 0' 23" the apparent altitude corrected for parallax. 
Hence, 5° 0' 23"-4° 49' 52"=10' 31", the refraction at the ap- 
parent altitude 5° 0' 14".* 

92. By these and similar methods, we could easily determine 
the refraction due to any elevation above the horizon, provided 
the refracting medium (the atmosphere) were always uniform. 
But this is not the fact : the refracting power of the atmosphere 
is altered by changes in density and temperature. f Hence in 
delicate observations, it is necessary to take into the account the 
state of the barometer and of the thermometer, the influence of 
the variations of each having been very carefully investigated, 
and rules having been given accordingly. With every precaution 
to insure accuracy, on account of the variable character of the 
refracting medium, the tables are not considered as entirely accu- 
rate to a greater distance from the zenith than 74° ; but almost all 
astronomical observations are made at a greater altitude than this. 

93. Since the whole amount of refraction near the horizon ex- 
ceeds 33', and the diameters of the sun and moon are severally 
less than this, these luminaries are in view both before they have 
actually risen and after they have set. 

94. The rapid increase of refraction near the horizon, is strik- 
ingly evinced by the oval figure which the sun assumes when 
near the horizon, and which is seen to the greatest advantage 
when light clouds enable us to view the solar disk. Were all 

* Gregory's Ast. p. 65. 

t It is said that the effects of humidity are insensible ; for the most accurate 
experiments seem to prove that watery vapor diminishes the density of air in the 
same ratio as its own refractive power is greater than that of air. (New Encyc. 
Brit. Ill, 762.) 



36 THE EARTH. 

parts of the sun equally raised by refraction, there would be no 
change of figure ; but since the lower side is more refracted than 
the upper, the effect is to shorten the vertical diameter and thus 
to give the disk an oval form. This effect is particularly remark- 
able when the sun, at his rising or setting, is observed from the 
top of a mountain, or at an elevation near the sea shore ; for in 
such situations the rays of light make a greater angle than or- 
dinary with a perpendicular to the refracting medium, and the 
amount of refraction is proportionally greater. In some cases of 
this kind, the shortening of the vertical diameter of the sun has 
been observed to amount to 6', or about one fifth of the whole.* 

95. The apparent enlargement of the sun and moon in the hori- 
zon, arises from an optical illusion. These bodies in fact are 
not seen under so great an angle when in the horizon, as when on 
the meridian, for they are nearer to us in the latter case than in 
the former. The distance of the sun is indeed so great that it 
makes very little difference in his apparent diameter, whether he 
is viewed in the horizon or on the meridian ; but with the moon 
the case is otherwise ; its angular diameter, when measured with 
instruments, is perceptibly larger at the time of its culmination. 
Why then do the sun and moon appear so much larger when near 
the horizon ? It is owing to that general law, explained in optics, 
by which we judge of the magnitudes of distant objects, not 
merely by the angle they subtend at the eye, but also by our im- 
pressions respecting their distance, allowing, under a given angle, 
a greater magnitude as we imagine the distance of a body to be 
greater. Now, on account of the numerous objects usually in 
sight between us and the sun, when on the horizon, he appears 
much further removed from us than when on the meridian, and 
we assign to him a proportionally greater magnitude. If we view 
the sun, in the two positions, through smoked glass, no such dif- 
ference of size is observed, for here no objects are seen but the 
sun himself. 



* In extreme cold weather, this shortening of the sun's vertical diameter sometimes 
exceeds this amount. 



TWILIGHT. 37 



TWILIGHT. 



96. Twilight also is another phenomenon depending upon the 
agency of the earth's atmosphere. It is due partly to refraction 
and partly to reflexion, but mostly the latter. While the sun 
is within 18° of the horizon, before it rises or after it sets, some 
portion of its light is conveyed to us by means of numerous re- 
flections from the atmosphere. Let AB (Fig. 10,) be the horizon 

Fig. 10. 




of the spectator at A, and let SS be a ray of light from the sun 
when it is two or three degrees below the horizon. Then to 
the observer at A, the segment of the atmosphere ABS would be 
illuminated. To a spectator at C, whose horizon was CD, the 
small segment SD# would be the twilight ; while, at E, the twi- 
light would disappear altogether. 

97. At the equator, where the circles of daily motion are per- 
pendicular to the horizon, the sun descends through 18° in an 
hour and twelve minutes (}f = l}h.), and the light of day there- 
fore declines rapidly, and as rapidly advances after daybreak in the 
morning. At the pole, a constant twilight is enjoyed while the sun 
is within 18° of the horizon, occupying nearly two thirds of the 
half year when the direct light of the sun is withdrawn, so that 
the progress from continual day to constant night is exceedingly 
gradual. To the inhabitants of an oblique sphere, the twilight 
is longer in proportion as the place is nearer the elevated pole. 

98. Were it not for the power the atmosphere has of dispersing 



38 THE EARTH. 

the solar light, and scattering it in various directions, no objects 
would be visible to us out of direct sunshine ; every shadow of a 
passing cloud would be pitchy darkness ; the stars would be visi- 
ble all day, and every apartment into which the sun had not di- 
rect admission, would be involved in the obscurity of night. This 
scattering action of the atmosphere on the solar light, is greatly 
increased by the irregularity of temperature caused by the sun, 
which throws the atmosphere into a constant state of undulation, 
and by thus bringing together masses of air of different tempera- 
tures, produces partial reflections and refractions at their common 
boundaries, by which means much light is turned aside from the 
direct course, and diverted to the purposes of general illumination.* 
In the upper regions of the atmosphere, as on the tops of very 
high mountains, where the air is too much rarefied to reflect much 
light, the sky assumes a black appearance, and stars become visi- 
ble in the day time. 



CHAPTER IV. 

OF TIME. 

99. Time is a measured portion of indefinite duration. 

Any event may be taken as a measure of time, which divides 
a portion of duration into equal parts ; as the pulsations of the 
wrist, the vibrations of a pendulum, or the passage of sand from 
one vessel into another, as in the hour-glass. 

100. The great standard of time is the period of the revolution 
of the earth on its axis, which, by the most exact observations, is 
found to be always the same. The time of the earth's revolution 
on its axis is called a sidereal day, and is determined by the revo- 
lution of a star from the instant it crosses the meridian, until it 
comes round to the meridian again. This interval being called a 

* Herschel. 



TIME. 39 

sidereal day, it is divided into 24 sidereal hours. Observations 
taken upon numerous stars, in different ages of the world, show 
that they all perform their diurnal revolutions in the same time, 
and that their motion during any part of the revolution is per- 
fectly uniform. 

101. Solar time is reckoned by the apparent revolution of the 

sun, from the meridian round to the same meridian again. Were 

the sun stationary in the heavens, like a fixed star, the time of its 

apparent revolution would be equal to the revolution of the earth 

on its axis, and the solar and the sidereal days would be equal. 

But since the sun passes from west to east, through 360° in 365£ 

days, it moves eastward nearly 1° a day, (59' 8".3). While, 

therefore, the earth is turning round on its axis, the sun is moving 

in the same direction, so that when we have come round under 

the same celestial meridian from which we started, we do not 

find the sun there, but he has moved eastward nearly a degree, 

and the earth must perform so much more than one complete 

revolution, in order to come under the sun again. Now since a 

place on the earth gains 359° in 24 hours, how long will it take 

to gain 1° ? 

24 
359 : 24 : : 1 : — =4 m nearly. 
359 J 

Hence the solar day is about 4 minutes longer than the sidereal ; 
and if we were to reckon the sidereal day 24 hours, we should 
reckon the solar day 24h. 4m. To suit the purposes of society at 
large, however, it is found most convenient to reckon the solar day 
24 hours, and to throw the fraction into the sidereal day. Then, 

24h. 4m. : 24 : : 24 : 23h. 56m. (23h. 56 m 4 S .09) = the length 
of a sidereal day. 

102. The solar days, however, do not always differ from the 
sidereal by precisely the same fraction, since the increments of 
right ascension, (Art. 37,) which measure the difference between 
a sidereal and a solar day, are not equal to each other. Apparent 
time, is time reckoned by the revolutions of the sun from the 
meridian to the meridian again. These intervals being unequal, 
of course the apparent solar days are unequal to each other. 



40 THE EARTH. 

103. Mean time, is time reckoned by the average length of all 
the solar days throughout the year. This is the period which con- 
stitutes the civil day of 24 hours, beginning when the sun is on 
the lower meridian, namely, at 12 o'clock at night, and counted 
by 12 hours from the lower to the upper culmination, and from 
the upper to the lower. The astronomical day is the apparent 
solar day counted through the whole 24 hours, instead of by pe- 
riods of 12 hours each, and begins at noon. Thus 10 days and 
14 hours of astronomical time, would be 11 days and 2 hours of 
civil time. 

104. Clocks are usually regulated so as to indicate mean solar 
time ; yet as this is an artificial period, not marked off, like the 
sidereal day, by any natural event, it is necessary to know how 
much is to be added to or subtracted from the apparent solar 
time, in order to give the corresponding mean time. The inter- 
val by which apparent time differs from mean time, is called the 
equation of time. If a clock were constructed (as it may be) so 
as to keep exactly with the sun, going faster or slower according 
as the increments of right ascension were greater or smaller, and 
another clock were regulated to mean time, then the difference 
of the two clocks, at any period, would be the equation of time 
for that moment. If the apparent clock were faster than the 
mean, then the equation of time must be subtracted ; but if the 
apparent clock were slower than the mean, then the equation of 
time must be added, to give the mean time. The two clocks 
would differ most about the 3d of November, when the apparent 
time is 16£ m greater than the mean (16 m 17 s ). But, since appa- 
rent time is sometimes greater and sometimes less than mean 
time, the two must obviously be sometimes equal to each other. 
This is in fact the case four times a year, namely, April 15th, 
June 15th, September 1st, and December 22d. These epochs, 
however, do not remain constant ; for, on account of the change 
in the position of the perihelion, or the point where the earth is 
nearest the sun, (which shifts its place from west to east about 
12" a year,) the period when the sun's motions are most rapid, as 
well as that when they are slowest, will occur at different parts of 
the year. The change is indeed exceedingly small in a single 



TIME. 



41 



year ; but in the progress of ages, the time of year when the sun's 
motion in its orbit is most accelerated, will not be, as at present, on 
the first of January, but may fall on the first of March, June, or 
any other day of the year, and the amount of the equation of 
time is obviously affected by the sun's distance from its perihelion, 
since the sun moves most rapidly when nearest the perihelion, and 
slowest when furthest from that point. 

105. The inequality of the solar days depends on two causes, the 
unequal motion of the earth in its orbit, and the inclination of the 
equator to the ecliptic. 

First, on account of the eccentricity* of the earth's orbit, the 
earth actually moves faster from the autumnal to the vernal equi- 
nox, than from the vernal to the autumnal, the difference of the 
two periods being about eight days (7d. 17h. 17m.) Thus, let 

Fig. 11. 




AEB (Fig. 11,) represent the earth's orbit, S being the place of 



* The exact figure of the earth's orbit will be more particularly shown hereafter. 
All that the student requires to know, in order to understand the present subject, 

6 



42 THE EARTH. 

the sun, A the perihelion, or nearest distance of the earth from 
the sun, B the aphelion, or greatest distance, and E, E', E", posi- 
tions of the earth in different points of its orbit. The place of 
the earth among the signs is the part of the heavens to which it 
would be referred if seen from the sun ; and the place of the sun 
is the part of the heavens to which it is referred as seen from the 
earth. Thus, when the earth is at E, it is said to be in Aries ; 
and as it moves from E through E' to A, its path in the heavens 
is through Aries, Taurus, Gemini, &c. Meanwhile the sun takes 
its place successively in Libra, Scorpio, Sagittarius, &c. Now, 
as will be shown more fully hereafter, the earth moves faster 
when proceeding from Aries through its perihelion to Libra, than 
from Libra through its aphelion to Aries, and, consequently, de- 
scribes the half of its apparent orbit in the heavens, T, £o, ^*, 
sooner than the half =£=, V3, T- The line of the apsides, that is, 
the major axis of the ellipse, is so situated at present, that the 
perihelion is in the sign Cancer, nearly 100° (99° 30' 5") from the 
vernal equinox. The earth passes through it about the first of 
January, and then its velocity is the greatest in the whole year, 
being always greater as the distance is less, the angular velocity 
being inversely as the square of the distance, as will be shown by 
and by. 

106. But differences of time are not reckoned on the eclip- 
tic, but on the equinoctial ; for the ecliptic being oblique to the 
meridian in the diurnal motion, and cutting it at different angles at 
different times, equal portions will not pass under the meridian in 
equal times, and therefore such portions could not be employed, as 
they are in the equinoctial, as measures of time. If therefore the 
sun moved uniformly in his orbit, so as to make the daily incre- 
ments of longitude equal, still the corresponding arcs of right as- 
cension, which determine the lengths of the solar day, would be 
unequal. Let us start from the equinox, from which both longi- 
tude and right ascension are reckoned, the former on the ecliptic, 



is that the earth's orbit is an ellipse, and that the earth's real motion, and conse- 
quently the sun's apparent motion, is greater in proportion as the earth is nearer 
the sun. 



TIME. 



43 



the latter on the equinoctial. Suppose the sun has described 70° 
of longitude ; then to ascertain the corresponding arc of right as- 
cension, we let a meridian pass through the sun : the point where 
it cuts the equator gives the sun's right ascension. Now since the 
ecliptic makes an acute angle with the meridian, while the equi- 
noctial makes a right angle with it, consequently the arc of longi- 
tude is greater than the arc of right ascension. The difference, 
however, grows constantly less and less as we approach the tropic, 
as the angle made between the ecliptic and the meridian constantly 
increases, until, when we reach the tropic, the meridian is at right 
angles to both circles, and the longitude and right ascension each 
equals 90°, and they are of course equal to each other. Beyond 
this, from the tropic to the other equinox, the arc of the ecliptic 
intercepted between the meridian and the autumnal equinox being 
greater than the corresponding arc of the equinoctial, of course 
its supplement, which measures the longitude, is less than the sup- 
plement of the corresponding arc of the equator which measures 
the right ascension. At the autumnal equinox again, the right 
ascension and longitude become equal. In a similar manner we 
might show that the daily increments of longitude and right as- 
cension are unequal. 

In order to illustrate the foregoing points, let T — (Fig. 12,) 

Fig. 12. 



] 
/ si 


\ 

T 


£ \ 


- /v -r"^ 




^ V *^J( 


E'j 




E 



°p 



represent the equator, T T =*= the ecliptic, and PSE, PS'E', two 
meridians meeting the sun in S and S'. Then in the triangle TES, 



44 THE EARTH. 

the arc of longitude TS, is greater than TE, the corresponding 
arc of right ascension ; but towards the tropic the difference 
between the two arcs evidently grows less and less, until at T 
the arcs become equal, being each 90°. But, beyond the tropic, 
since TE'^ TS'^, are equal to each other, each being equal 
to 180°, and since S'^= is greater than E'^=, therefore TS' must 
be less than TE'. 

107. As the whole arc of right ascension reckoned from the 
first of Aries, does not keep uniform pace with the corresponding 
arc of longitude, so the daily increments of right ascension differ 
from those of longitude. If we suppose in the quadrant TT, 
points taken to mark the progress of the sun from day to day, and 
let meridians like PSE pass through these points, the arc of the 
ecliptic embraced between the meridians will be the daily incre- 
ments of longitude, while the corresponding parts of the equinoc- 
tial will be the daily increments of right ascension. Near T, the 
oblique direction in which the ecliptic cuts the meridian, will make 
the daily increments of longitude exceed those of right ascension ; 
but this advantage is diminished as we approach the tropic, where 
the ecliptic becomes less oblique, and finally parallel to the equi- 
noctial ; while the convergence of the meridians contributes still 
farther to lessen the ratios of the daily increments of longitude to 
those of right ascension. Hence, at first, the diurnal arcs of 
right ascension are less than those of longitude, but afterwards 
greater ; and they continue greater for a similar distance beyond 
the tropic. 

108. From the foregoing considerations it appears, that the 
diurnal arcs of right ascension, by which the difference between 
the sidereal and the solar days is measured, are unequal, on ac- 
count both of the unequal motion of the sun in his orbit, and of 
the inclination of his orbit to the equinoctial. 

109. As astronomical time commences when the sun is on the 
meridian, so sidereal time commences when the vernal equinox 
is on the meridian, and is also counted from to 24 hours. By 
3 o'clock, for instance, of sidereal time, we mean that it is three 



THE CALENDAR. 45 

hours since the vernal equinox crossed the meridian ; as we say it 
is 3 o'clock of astronomical or of civil time, when it is three hours 
since the sun crossed the meridian. v& 



THE CALENDAR. 

110. The astronomical year is the time in which the sun makes 
one revolution in the ecliptic, and consists of 365d. 5h. 48m. 5P.60. 
The civil year consists of 365 days. The difference is nearly 6 
hours, making one day in four years. 

111. The most ancient nations determined the number of days 
in the year by means of the stylus, a perpendicular rod which 
cast its shadow on a smooth plane, bearing a meridian line. The 
time when the shadow was shortest, would indicate the day of 
the summer solstice ; and the number of days which elapsed until 
the shadow returned to the same length again, would show the 
number of days in the year. This was found to be 365 whole 
days, and accordingly this period was adopted for the civil year. 
Such a difference, however, between the civil and astronomical 
years, at length threw all dates into confusion. For, if at first 
the summer solstice happened on the 21st of June, at the end of 
four years, the sun would not have reached the solstice until the 
22d of June, that is, it would have been behind its time. At the 
end of the next four years the solstice would fall on the 23d ; 
and in process of time it would fall successively on every day of 
the year. The same would be true of any other fixed date. 
Julius Caesar made the first correction of the calendar, by intro- 
ducing an intercalary day every fourth year, making February 
to consist of 29 instead of 28 days, and of course the whole year 
to consist of 366 days. This fourth year was denominated Bis- 
sextile,* It is also called Leap Year. 

112. But the true correction was not 6 hours, but 5h. 49m.; 
hence the intercalation was too great by 11 minutes. This small 
fraction would amount in 100 years to £ of a day, and in 1000 

* The sextus dies ante Kalendas being reckoned twice, (Bis). 



46 THE EARTH. 

years to more than 7 days. From the year 325 to 1582, it had 
in fact amounted to about 10 days; for it was known that in 325, 
the vernal equinox fell on the 21st of March, whereas, in 1582 it 
fell on the 11th. In order to restore the equinox to the same date, 
Pope Gregory XIII decreed, that the year should be brought for- 
ward ten days, by reckoning the 5th of October the 15th. In or- 
der to prevent the calendar from falling into confusion afterwards, 
the following rule was adopted : 

Every year whose number is not divisible by 4 without a re- 
mainder, consists of 365 days ; every year which is so divisible, but 
is not divisible by 100, of 366 ; every year divisible by 100 but not 
by 400, again of 365 ; and every year divisible by 400, of 366. 

Thus the year 1838, not being divisible by four, contains 365 days, 
while 1836 and 1840 are leap years. Yet to make every fourth 
year consist of 366 days would increase it too much by about £ 
of a day in 100 years ; therefore every hundredth year has only 
365 days. Thus 1800, although divisible by 4, was not a leap 
year, but a common year. But we have allowed a whole day 
in a hundred years, whereas we ought to have allowed only three 
fourths of a day. Hence, in 400 years we should allow a day too 
much, and therefore we let the 400th year remain a leap year. 
This rule involves an error of less than a day in 4237 years.* If 
the rule were extended by making every year divisible by 4,000 
(which would now consist of 366 days) to consist of 365 days, the 
error would not be more than one day in 100,000 years.f 

113. This reformation of the calendar was not adopted in Eng- 
land until 1752, by which time the error in the Julian calendar 
amounted to about 1 1 days. The year was brought forward, by 
reckoning the 3d of September the 14th. Previous to that time 
the year began the 25th of March; but it was now made to be- 
gin on the 1st of January, thus shortening the preceding year, 
1751, one quarter. J 






* Woodhouse, p. 674. t Herschel's Ast. p. 384. 

X Russia, and the Greek Church generally, adhere to the old style. In order to make 
the Russian dates correspond to ours, we must add to them 12 days. France and other 
Catholic countries, adopted the Gregorian calendar soon after it was promulgated. 



THE CALENDAR. 47 

114. As in the year 1582, the error in the Julian calendar 
amounted to 10 days, and increased by f of a day in a century, 
at present the correction is 12 days ; and the number of the year 
will differ by one with respect to dates between the 1st of Janu- 
ary and the 25th of March. 

Examples. General Washington was born Feb. 11, 1731, old 
style ; to what date does this correspond in new style ? 

As the date is the earlier part of the 18th century, the correc- 
tion is 1 1 days, which makes the birth day fall on the 22d of 
February; and since the year 1731 closed the 25th of March, 
while according to new style 1732 would have commenced on 
the preceding 1st of January ; therefore, the time required is Feb. 
22, 1732. It is usual, in such cases, to write both years, thus: 
Feb. 11, 1731-2, O. S. 

2. A great eclipse of the sun happened May 15th, 1836; to 
what date would this time correspond in old style 1 

Ans. May. 3d. 

115. The common year begins and ends on the same day of 
the week ; but leap year ends one day later in the week than it began. 

For 52x7=364 days; if therefore the year begins on Tues- 
day, for example, 364 days would complete 52 weeks, and one 
day would be left to begin another week, and the following year 
would begin on Wednesday. Hence, any day of the month is one 
day later in the week than the corresponding day of the preceding 
year. Thus, if the 16th of November, 1838, falls on Friday, 
the 16th of November, 1837, fell on Thursday, and will fall in 
1839 on Saturday. But if leap year begins on Sunday, it ends 
on Monday, and the following year begins on Tuesday ; while 
any given day of the month is two days later in the week than 
the corresponding date of the preceding year. 

116. Fortunately for astronomy, the confusion of dates involved 
in different calendars affects recorded observations but little. Re- 
markable eclipses, for example, can be calculated back for several 
thousand years, without any danger of mistaking the day of their 
occurrence ; and whenever any such eclipse is so interwoven with 
the account given by an ancient author of some historical event, 



48 THE EARTH. 



as to indicate precisely the interval of time between the eclipse 
and the event, and at the same time completely to identify the 
eclipse, that date is recovered and fixed forever.* 



CHAPTER V. 

OF ASTRONOMICAL INSTRUMENTS AND PROBLEMS FIGURE AND DEN- 
SITY OF THE EARTH. 

117. The most ancient astronomers employed no instruments 
for measuring angles, but acquired their knowledge of the heav- 
enly bodies by long continued and most attentive inspection with 
the naked eye. In the Alexandrian school, about 300 years before 
the Christian era, instruments began to be freely used, and thence- 
forward trigonometry lent a powerful aid to the science of astron- 
omy. Tycho Brahe, in the 16th century, formed a new era in 
practical astronomy, and carried the measurement of angles to 
10", — a degree of accuracy truly wonderful, considering that he 
had not the advantage of the telescope. By the application of 
the telescope to astronomical instruments, a far better defined view 
of objects was acquired, and a far greater degree of refinement 
was attainable. The astronomers royal of Great Britain perfected 
the art of observation, bringing the measurement of angles to 1", 
and the estimation of differences of time to T V of a second. Be- 
yond this degree of refinement it is supposed that we cannot 
advance, since unavoidable errors arising from the uncertainties 
of refraction, and the necessary imperfection of instruments, for- 
bid us to hope for a more accurate determination than this. But 
a little reflection will show us, that 1" on the limb of an astro- 
nomical instrument, must be a space exceedingly small. Suppose 
the circle, on which the angle is measured, be one foot in diameter. 



* An elaborate view of the Calendar may be found in Delambre's Astronomy, t. III. 
A useful table for finding the day of the week of any given date, is inserted in the 
American Almanac for 1832, p. 72. 



ASTRONOMICAL INSTRUMENTS. 49 

rp, 12x3.14159 T . , . , , io TT 

1 hen — — = T V men = space occupied by 1 . Hence 

360 

ro4eo =6Uo =space of V ' and 3^000 =space of J "- Suchmi » ute 

angles can be measured only by large circles. If, for example, 
a circle is 20 feet in diameter, a degree on its periphery would 
occupy a space 20 times as large as a degree on a circle of 1 foot. 
A degree therefore of the limb of such an instrument would 
occupy a space of 2 inches : one minute, 3V of an inch ; and one 
second, T¥ Vo °f an inch. 

118. But the actual divisions on the limb of an astronomical 
instrument never extend to seconds : in the smaller instruments 
they reach only to 10', and on the largest rarely lower than V. 
The subdivision of these spaces is carried on by means of the 
Vernier, which may be thus defined : 

A Vernier is a contrivancce attached to the graduated limb of 
an instrument, for the purpose of measuring aliquot parts of the 
smallest spaces, into which the instrument is divided. 

The vernier is usually a narrow zone of metal, which is made 
to slide on the graduated limb. Its divisions correspond to those 
on the limb, except that they are a little larger,* one tenth, for 
example, so that ten divisions on the vernier would equal eleven 
on the limb. Suppose now that our instrument is graduated to 
degrees only, but the altitude of a certain star is found to be 40° 
and a fraction, or 40° +x. In order to estimate the amount of this 
fraction, we bring the zero point of the vernier to coincide with 
the point which indicates the exact altitude, or 40° +x. We then 
look along the vernier until we find where one of its divisions 
coincides with one of the divisions of the limb. Let this be at the 
fourth division of the vernier. In four divisions, therefore, the ver- 
nier has gained upon the divisions of the limb, a space equal to x ; 
and since, in the case supposed, it gains T V of a degree, or 6' at each 
division, the entire gain is 24', and the arc in question is 40° 24'. 

119. As the vernier is used in the barometer, where its applica- 

* In the more modern instruments the divisions of the vernier are smaller than those 
of the limb. 

7 



50 



THE EARTH. 



tion is more easily seen than in astronomical instruments, while the 
principle is the same in both cases, let us 
see how it is applied to measure the ex- 
act height of a column of mercury. Let 
AB (Fig. 13,) represent the upper part 
of a barometer, the level of the mercury 
being at C, namely, at 30.3 inches, and 
nearly another tenth. The vernier being 
brought (by a screw which is usually at- 
tached to it) to coincide with the surface 
of the mercury, we look along down the 
scale, until we find that the coincidence 
is at the 8th division of the vernier. 
Now as the vernier gains T \ of J_== T i t 
of an inch at each division upward, it of 
course gains T | F in eight divisions. The fractional quantity, there- 
fore, is .08 of an inch, and the height of the mercury is 30.38. If 
the divisions of the vernier were such, that each gained ¥ V (when 
60 on the vernier would equal 61 on the limb) on a limb divided 
into degrees, we could at once take off minutes ; and were the limb 
graduated to minutes, we could in a similar way read off seconds. 





Fig 


13. 






A f 






j— 31 


c 










fill 


l 




— 




'ii: 




2 


— 


— 








S 
4 


— 




— 30 






5 
6 

7 
8 


— 










9 

1(1 


— 


- 














—29 


Bill 


ill III 











120. The instruments most used for astronomical observations, 
are the Transit Instrument, the Astronomical Clock, the Mural 
Circle, and the Sextant. A large portion of all the observations, 
made in an astronomical observatory, are taken on the meridian. 
When a heavenly body is on the meridian, being at its highest 
point above the horizon, it is then least affected by refraction and 
parallax ; its zenith distance (from which its altitude and decli- 
nation are easily derived) is readily estimated ; and its right as- 
cension may be very conveniently and accurately determined by 
means of the astronomical clock. Having the right ascension 
and declination of a heavenly body, various other particulars re- 
specting its position may be found, as we shall see hereafter, by 
the aid of spherical trigonometry. Let us then first turn our at- 
tention to the instruments employed for determining the right 
ascension and declination. They are the Transit Instrument, the 
Astronomical Clock, and the Mural Circle. 



ASTRONOMICAL INSTRUMENTS. 



51 



121. The Transit Instrument is a telescope, which is fixed 
permanently in the meridian, and moves only in that plane. It 
rests on a horizontal axis, which consists of two hollow cones 
applied base to base, a form uniting lightness and strength. The 
two ends of the axis rest on two firm supports, as pillars of stone, 
for example, usually built up from the ground, and so related to 
the building as to be as free as possible from all agitation. In 
figure 14, AD represents the telescope, E, W, massive stone pillars 
supporting the horizontal axis, beneath which is seen a spirit level, 
(which is used to bring the axis to a horizontal position,) and n a 
vertical circle graduated into degrees and minutes. This circle 
serves the purpose of placing the instrument at any required alti- 
tude or distance from the zenith, and of course for determining 
altitudes and zenith distances. 

Fig. 14. 




122. Various methods are described in works on practical as- 
tronomy, for placing the Transit Instrument accurately in the 
meridian. The following method by observations on the pole 
star, may serve as an example. If the instrument be directed 



52 



THE EARTH. 



towards the north star, and so adjusted that the star Alioth (the 
first in the tail of the Great Bear) and the pole star are both in 
the same vertical circle, the former below the pole and the latter 
above it, the instrument will be nearly in the plane of the meridian. 
To adjust it more exactly, compare the time occupied by the pole 
star in passing from its upper to its lower culmination, with the 
time of passing from its lower to its upper culmination. These 
two intervals ought to be precisely equal ; and if they are so, the 
iustrument is truly placed in the meridian ; but if they are not 
equal, the position of the instrument must be shifted until they 
become exactly equal. 



123. The line of collimation of a telescope, is a line joining the 
center of the object glass with the center of the eye glass. When 
the transit instrument is properly adjusted, this line, as the instru- 
ment is turned on its axis, moves in the plane of the meridian. 
Having, by means of the vertical circle n, set the instrument at 
the known altitude or zenith distance of any star, upon which we 
wish to make observations, we wait until the star enters the field 
of the telescope, and note the exact instant when it crosses the 
vertical wire in the center of the field, which wire marks the true 
plane of the meridian. Usually, however, there are placed in the 
focus of the eye glass five parallel wires or threads, two on each 
side of the central wire, and all 
at equal distances from each 
other, as is represented in the 
following diagram. The time 
of arriving at each of the wires 
being noted, and all the times 
added together and divided by 
the number of observations, the 
result gives the instant of cross- 
ing the central wire. 

124. The Astronomical Clock 
is the constant companion of the 
Transit Instrument. This clock is so regulated as to keep exact 
pace with the stars, and of course with the revolution of the earth 







ASTRONOMICAL INSTRUMENTS. 53 

on its axis ; that is, it is regulated to sidereal time. It measures 
the progress of a star, indicating an hour for every 15°, and 24 
hours for the whole period of the revolution of the star. Sidereal 
time, it will be recollected, commences when the vernal equinox 
is on the meridian, just as solar time commences when the sun is 
on the meridian. Hence, the hour by the sidereal clock has no 
correspondence with the hour of the day, but simply indicates 
how long it is since the equinoctial point crossed the meridian. 
For example, the clock of an observatory points to 3h. 20m. ; this 
may be in the morning, at noon, or any other time of the day, since 
it merely shows that it is 3h. 20m. since the equinox was on the 
meridian. Hence, when a star is on the meridian, the clock 
itself shows its right ascension ; and the interval of time between 
the arrival of any two stars upon the meridian, is the measure of 
their difference of right ascension. 

125. Astronomical clocks are made of the best workmanship, 
with a compensation pendulum, and every other advantage which 
can promote their regularity. The Transit Instrument itself, 
when once accurately placed in the meridian, affords the means 
of testing the correctness of the clock, since one revolution of a 
star from the meridian to the meridian again, ought to correspond 
to exactly 24 hours by the clock, and to continue the same from 
day to day ; and the right ascension of various stars, as they cross 
the meridian, ought to be such by the clock as they are given in 
the tables, where they are stated according to the most accurate 
determinations of astronomers. Or by taking the difference of 
right ascension of any two stars on successive days, it will be seen 
whether the going of the clock is uniform for that part of the 
day ; and by taking the right ascension of different pairs of stars, 
we may learn the rate of the clock at various parts of the day. 
We thus learn, not only whether the clock accurately measures 
the length of the sidereal day, but also whether it goes uniformly 
from hour to hour. 

Although astronomical clocks have been brought to a great de- 
gree of perfection, so as to vary hardly a second for many months, 
yet none are absolutely perfect, and most are so far from it as to 
require to be corrected by means of the Transit Instrument every 



54 THE EARTH. 

few days. Indeed, for the nicest observations, it is usual not to 
attempt to bring the clock to an absolute state of correctness, but 
after bringing it as near to such a state as can conveniently be 
done, to ascertain how much it gains or loses in a day ; that is, to 
ascertain its rate of going, and to make allowance accordingly. 

126. The vertical circle (n, Fig. 14,) usually connected with 
the Transit Instrument, affords the means of measuring arcs on 
the meridian, as meridian altitudes, zenith distances, and decli- 
nations ; but as the circle must necessarily be small, and there- 
fore incapable of measuring very minute angles, the Mural Cir- 
cle is usually employed for measuring arcs of the meridian. The 
Mural Circle is a graduated circle, usually of very large size, fixed 
permanently in the plane of the meridian, and attached firmly to 
a perpendicular wall. It is made of large size, sometimes 1 1 feet 
in diameter, in order that very small angles may be measured on 
its limb ; and it is attached to a massive wall of solid masonry in 
order to insure perfect steadiness, a point the more difficult to 
attain in proportion as the instrument is heavier. The annexed 
diagram represents a Mural Circle fixed to its wall and ready for 
observations. It will be seen that every expedient is employed 
to give the instrument firmness of parts and steadiness of position. 
Its radii are composed of hollow cones, uniting lightness and 
strength, and its telescope revolves on a large horizontal axis, 
fixed as firmly as possible in a solid wall. The graduations are 
made on the outer rim of the instrument, and are read off by six 
microscopes (called leading microscopes) attached to the wall, one 
of which is represented at A, and the places of the five others 
are marked by the letters B, C, D, E, F. Six are used, in order 
that by taking the mean of such a number of readings, a higher 
degree of accuracy may be insured, than could be obtained by a 
single reading. In a circle of six feet diameter, like that repre- 
sented in the figure, the divisions may be easily carried to five 
minutes each. The microscope (which is of the variety called 
compound microscope) forms an enlarged image of each of these 
divisions in the focus of the eye glass. With it is combined the 
principle of the micrometer. This is effected by placing in the 
focus a delicate wire, which may be moved by means of a screw 



ASTRONOMICAL INSTRUMENTS. 

Fig. 16. 



55 




in a direction parallel to the divisions of the limb, and which is so 
adjusted to the screw as to move over the whole magnified space 
of five minutes by five revolutions of the screw. Of course one 
revolution of the screw measures one minute. Moreover, if the 
screw itself is made to carry an index attached to its axis and re- 
volving with it over a disk graduated into sixty equal parts, then 
the space measured by moving the index over one of these parts, 
will be one second. 

We have been thus minute in the description of this instrument, 
in order to give the learner some idea of the vast labor and great 
patience demanded of practical astronomers, in order to obtain 
measurements of such extreme accuracy as those to which they 
aspire. 

On account of the great dimensions of this circle, and the ex- 
pense attending it, as well as the difficulty of supporting it firmly, 
sometimes only one fourth of it is employed, constituting the Mu- 
ral Quadrant. This instrument has the disadvantage, however, 



56 



THE EARTH. 



of being applicable to only one hemisphere at a time, either the 
northern or the southern, according as it is fixed to the eastern 
or the western side of the wall. 



127. We have before shown (Art. 124,) the method of finding 
the right ascension of a star by means of the Transit Instrument 
and the clock. The declination may be obtained by means of the 
mural circle in several different ways, our object being always to 
find the distance of the star, when on the meridian, from the equa- 
tor (Art. 37.) First, the declination may be found from the me- 
ridian altitude. Let S (Fig. 17,) be the place of a star when 
on the meridian. Then its meridian altitude will be SH, which 
will best be found by taking its ze- 
nith distance ZS, of which it is the 
complement. From SH, subtract EH, 
the elevation of the equator, which 
equals the co-latitude of the place of 
observation, (Art. 44,) and the remain- 
der SE is the declination. Or if the 
star is nearer the horizon than the 
equator is, as at $', subtract its me- 
ridian altitude from the co-latitude, for 
the declination. Secondly, the declination may be found from 
the north polar distance, of which it is the complement. Thus 
from P to E is 90°. Therefore, PE-PS=90°-PS=SE=the 
declination. The height of the pole P is always known when the 
latitude of the place is known, being equal to the latitude. 

128. The astronomical instruments already described are adapt- 
ed to taking observations on the meridian only ; but we some- 
times require to know the altitude of a celestial body when it is 
not on the meridian, and its azimuth, or distance from the meridian 
measured on the horizon ; and also the angular distance between 
two points on any part of the celestial sphere. An instrument 
especially designed to measure altitudes and azimuths, is called an 
Altitude and Azimuth Instrument, whatever may be its particular 
form. When a point is on the horizon its distance from the me- 
ridian, or its azimuth, may be taken by the common surveyor's 




ASTRONOMICAL INSTRUMENTS. 



57 



compass, the direction of the meridian being determined by the 
needle ; but when the object, as a star, is not on the horizon, its 
azimuth, it must be remembered, is the arc of the horizon from 
the meridian to a vertical circle passing through the star (Art. 27) ; 
at whatever different altitudes, therefore, two stars may be, and 
however the plane which passes through them may be inclined to 
the horizon, still it is their angular distance measured on the hori- 
zon which determines their difference of azimuth. Figure 18 rep- 
resents an Altitude and Azimuth Instrument, several of the usual 
appendages and subordinate contrivances being omitted for the 
sake of distinctness and simplicity. Here abc is the vertical or 
altitude circle, and EFG the horizontal or azimuth circle ; AB is a 

Fig. 18. 




telescope mounted on a horizontal axis and capable of two mo- 
tions, one in altitude parallel to the circle abc, and the other in 
azimuth parallel to EFG. Hence it can be easily brought to bear 
upon any object. At m, under the eye glass of the telescope, is a 
small mirror placed at an angle of 45° with the axis of the tele- 
scope, by means of which the image of the object is reflected up- 
wards, so as to be conveniently presented to the eye of the ob- 

8 



58 



THE EARTH. 



server. At d is represented a tangent screw, by which a slow 
motion is given to the telescope at c. At h and g are seen two 
spirit levels, at right angles to each other, which show when the 
azimuth circle is truly horizontal. The instrument is supported 
on a tripod, for the sake of greater steadiness, each foot being fur- 
nished with a screw for levelling. 

129. The sextant* is one of the most useful instruments, both 
to the astronomer and the navigator, and will therefore merit 
particular attention. In figure 1 9, ABL represents the plane of 
the instrument, LG and N, two small mirrors, and T, a small tele- 
scope. The line LGI represents a movable arm, or radius, which 
carries an index at I. The radius turns on a pivot in the center 

Fig. 19. 




of LG, and the index moves on a graduated arc BA. LG is called 
the Index Glass, and N the Horizon Glass. The under part only 
of the horizon glass is coated with quicksilver, the upper part be- 
ing left transparent, as in n ; so that while one object is seen 
through the upper part of n by direct vision, another may be seen 
through the lower part by reflexion from the two mirrors. The 
instrument is so contrived, that when the index is moved up to A, 
where the zero point is placed, or the graduation begins, the two 
reflectors LG and N are exactly parallel to each other, the index 
glass being then in the position Ig. In this position of the mir- 
rors, if the eye at E look through the telescope, T, so pointed as to 



* See Fig. 15, Mason's Supplement. 



ASTRONOMICAL INSTRUMENTS. 59 

see the star S through the transparent part of the horizon glass, it 
will see the same star, in the same place reflected from the sil- 
vered part ; for the star (or any similar object) is at such a dis- 
tance that the rays of light which strike upon the index glass LG 
are parallel to those which enter the eye directly. Therefore the 
angle of incidence bcN being equal to the angle of reflexion at 
cNE, the ray b will be made, by reflexion, to coincide with the 
ray a, and exhibit the object at the same place. Now, suppose we 
wish to measure the angular distance between two bodies, as the 
moon and a star, and let the star be at S and the moon at M. 
The telescope being still directed to S, turn the index arm LI from 
A towards B until the image of the moon is brought down to S, 
its lower limb just touching S. By a principle in optics, the an- 
gular distance which the image of the moon passes over, is twice 
that of the mirror LG. But the mirror has passed over the grad- 
uated arc AI ; therefore double that arc is the angular distance 
between the star and the moon's lower limb. If we then bring 
the upper limb into contact with the star, the sum of both obser- 
vations, divided by 2, will give the angular distance between the 
star and the moon's center. As each degree on the limb AB meas- 
ures two degrees of angular distance, hence the divisions for sin- 
gle degrees are in fact only half a degree asunder ; and a sextant, 
or the sixth part of the circle, measures an angular distance of 
120°. The upper and lower points in the disk of the sun or of 
the moon may be considered as two separate objects, whose dis- 
tance from each other may be taken in a similar manner, and thus 
their apparent diameters at any time be determined. We may 
select our points of observation either in a vertical, or in a hori- 
zontal direction. 

130. If we make a star, or the limb of the sun or moon, one 
,of the objects, and the point in the horizon directly beneath, the 
other, we thus obtain the altitude of the object. In this observa- 
tion, the horizon is viewed through the transparent part of the 
horizon glass. At sea, where the horizon is usually well defined, 
the horizon itself may be used for taking altitudes; but on land, 
inequalities of the earth's surface, oblige us to have recourse to 
an artificial horizon. This, in its simple state, is a basin of either 



60 THE EARTH. 

water or quicksilver. By this means we see the image of the 
sun (or other body) just as far below the horizon as it is in reality 
above it. Hence, if we turn the index glass until the limb of the 
sun, as reflected from it, is brought into contact with the image 
seen in the artificial horizon, we obtain double the altitude.* 

The sextant must be held in such a manner, that its plane shall 
pass through the plane of the two objects. It must be held 
therefore in a vertical plane in taking altitudes, and in a horizontal 
plane in taking the horizontal diameters of the sun and moon. 
Holding the instrument in the true plane of the two bodies, whose 
angular distance is measured, is indeed the most difficult part of 
the operation. 

The peculiar value of the sextant consists in this, that the ob- 
servations taken with it are not affected by any motion in the 
observer ; hence it is the chief instrument used for angular meas- 
urements at sea. 

131. Examples illustrating the use of the Sextant. 
Ex. 1. Alt. 0's lower limb, . . 49° 10' 00" 

O's semi-diameter, . . 15 51 

Subtract Refraction, 

Add Parallax, 

True altitude O's center, . 49° 25' 08" 

Ex. 2. With the Artificial Horizon. 
Altitude of o's upper limb above the image in the artificial ho- 
rizon, 100° 2' 47". 

True altitude, 50° 01' 23."5 

O's semi-diameter, . . . . 00 15 50. 

Refraction, 

Parallax, 

True altitude of O's center, ... 49° 44' 50."5 

* Woodhouse's Ast. p. 774. 



49° 


25' 


51" 


00 


00 


49 


49° 


25' 


02" 


00 


00 


06 



49° 


45' 


33. 


''5 


00 


00 


48. 




49° 


44' 


45. 


"5 


00 


00 


05. 






ASTRONOMICAL PROBLEMS. 61 

ASTRONOMICAL PROBLEMS.* 

132. Given the sun's Right Ascension and Declination, to find 
his Longitude and the Obliquity of the Ecliptic. 

Let PCP' (Fig. 20,) represent the celestial meridian that passes 
through the first of Cancer and Capricorn, (the solstitial colure,) 
PP' the axis of the sphere, EQ the equator, E'C the ecliptic, and 
PSP' the declination circle (Art. 
37,) passing through the sun S ; 
then ARS is a right angle, and in 
the right angled spherical triangle 
ARS, are given the right ascension 
AR (Art. 37,) and the declination 
RS, to find the longitude AS and 
the obliquity SAR. 

As longitude and right ascension 
are measured from A, the first point 

of Aries, in the direction AS of the signs, quite round the globe, 
when, of the four quantities mentioned in the problem, the obliquity 
and the declination are given to find the others, we must know 
whether the sun is north, or whether it is south of the equator, the 
longitude being in the one case AS, and in the other, instead of 
AS', it is 360— AS', that is, the supplement of AS'. We must 
also know on which side of the tropic the sun is, for the sun in 
passing from one of the tropics to the equinox, passes through the 
same degrees of declination as it had gone through in ascending 
from the other equinox to the tropic, although its longitude and 
right ascension go on continually increasing. From the 21st of 
March to the 21st of June, while describing the first quadrant 
from the vernal equinox, the declination is north and increasing ; 
north but decreasing, in the second quadrant, until the 23d of 
September ; south and increasing in the third quadrant, until the 
21st of December; and finally, in the fourth quadrant, south but 
decreasing until the 21st of March. 

Ex. I. On the 17th of May, the sun's Right Ascension was 
53° 38', and his Declination 19° 15' 57": required his Longitude 
and the Obliquity of the Ecliptic. 

* Young's Spherical Trigonometry, p. 136. Vince's Complete System, Vol. I. 



62 THE EARTH. 

Applying Napier's rule* to the right angled triangle, ARS, we 
have 

1. Rad. cos. AS=cos. AR cos. RS. 

2. Rad. sin. AR=tan. RS cot. A. -.cot. A= £^^A?:. 

tan. RS 

Hence the computation for AS and A is as follows : 

For the Longitude AS. For the Obliquity A. 



cos.AR 53° 38' 00" 9.7730185 
cos.RS 19 15 57 9.9749710 



cos.AS 55 57 43 9.7479895 



sin AR 9.9059247 

tan. RS, ar. com. 0.4565209 



cot. A 23° 27' 50i" 10.3624456 



Ex. 2. On the 31st of March, 1816, the sun's Declination was 
observed at Greenwich to be 4° 13' 31$": required his Right 
Ascension, the obliquity of the ecliptic being 23° 27' 51". 

Ans. 9° 47' 59". 

Ex. 3. What was the sun's Longitude on the 28 th of Novem- 

* The student is supposed to be acquainted with Spherical Trigonometry ; but to re- 
fresh his memory, we may insert a remark or two. 

It will be recollected that in Napier's rule for the solution of a right angled spherical 
triangle, by means of the Five Circular Parts, we proceed as follows. 

In a right angled spherical triangle we are to recognize but five parts, viz. the three 
sides and the two oblique angles. If we take any one of these as a middle part, the 
two which lie next to it on each side will be adjacent parts. Thus, (in Fig. 21,) taking 
A for a middle part, h and c will be the adjacent parts ; if we take c for the middle part, 
A and B will be the adjacent parts ; if we Fig. <n\ m 

take B for the middle part, c and a, will be 
the adjacent parts ; but if we take a for 
the middle part, then as the angle C is 
not considered as one of the circular parts, 
B and b are the adjacent parts ; and, last- 
ly, if b is the middle part, then the adja- 
cent parts are A and a. The two parts immediately beyond the adjacent parts on each 
side, still disregarding the right angle, are called the opposite parts ; thus if A is the 
middle part, the opposite parts are a and B. Napier's rule is as follows : 

Radius into the sine of the middle part, equals the product of the tangents of the 
adjacent extremes, or of the cosines of the opposite extremes. 

(The corresponding vowels are marked to aid the memory.) This rule is modified 
by using the complements of the two angles and the hypothenuse instead of the parts 
themselves. Thus instead of rad. X sin. A, we say rad. X cos. A, when A is the middle 
part ; and rad. X cos. AB, when the hypothenuse is the middle part. 

Examples. 1. In the right angled triangle ABC, are given the two perpendicular 
sides, viz. a=48° 24' 10", 6=59° 38' 27", to find the hypothenuse c. The hypothenuse 
being made the middle part, the other sides become the opposite parts, being separated 




ASTRONOMICAL PROBLEMS. 63 

ber, 1810, when his Declination was 21° 16' 4", and his Right 
Ascension, in time, 16h. 14m. 58.4s.? 

Ans. 245° 39' 10". 
Ex. 4. The sun's Longitude being 8s. 7° 40' 56", and the Ob- 
liquity 23° 27' 42 i", what was the Right Ascension in time? 

Ans. 16h. 23m. 34s. 

1 33. Given the sun's Declination to find the time of his Rising 
and Setting at any place whose latitude is known. 

Let PEP' (Fig. 22,) represent the meridian of the place, Z 
being the zenith, and HO the horizon ; and let LL' be the appa- 
rent path of the sun on the proposed 
day, cutting the horizon in S. Then 
the arc EZ will be the latitude of the 
place, and consequently EH, or its 
equal QO, will be the co-latitude, and 
this measures the angle OAQ ; also 
RS will be the sun's declination, and 
AR expressed in time will be the time 
of rising before 6 o'clock. For it is 
evident that it will be sunrise when 

the sun arrives at the horizon at S ; but PP' being an hour circle 
whose plane is perpendicular to the meridian, (and of course pro- 
jected into a straight line on the plane of projection,) the time the 
sun is passing from S to S' taken from the time of describing S'L, 
which is six hours, must be the time from midnight to sunrise. 
But the time of describing SS' is measured on the corresponding 
arc of the equinoctial AR. 

In the right angled triangle ARS, we have the declination RS, 
and the angle A to find AR. Therefore, 
Rad. xsin. AR=cot. A xtan. RS. 




from the middle part by the angles A and B. Hence, rad. cos. c=cos. a cos. b .-. cos. c= 
cos.*cos.^ 70o26 , 29 ,, 
rad. 

2. In the spherical triangle, right angled at C, are given two perpendicular sides, 
viz. a=116° 30' 43", 6=29° 41' 32", to find the angle A. 

Here, the required angle is adjacent to one of the given parts, viz. 6, which make 
the middle part. Then, 

Rad.xsin. 6=cot A tan. a .-.cot. A = rad - Xsin - 6 =76 o 7 , 14 * 

tan. a. 



64 



THE EARTH. 



Ex. 1. Required the time of sunrise at latitude 52° 13' N. 
when the sun's declination is 23° 28'. 



Rad 

Cot. A or tan. 52° 13' 
Tan. RS= 23° 28' 

Sin. 34° 03 211" 

4# 

2h, 16' 13"25 ;// 
6 



10. 

10.1105786 
9.6376106 

9.7481892 



3h. 43' 46" 35"'= the time after midnight, and of 
course the time of rising. 

Ex. 2. Required the time of sunrise at latitude 57° 2' 54" N. 
when the sun's declination is 23° 28' N. 

Ans. 3h. 11m. 49s. 
Ex. 3. How long is the sun above the horizon in latitude 58° 
12' N. when his declination is 18° 40' S. ? 

Ans. 7h. 35m. 52s. 



134. Given the Latitude of the place, and the Declination of a 
heavenly body, to determine its Altitude and Azimuth when on the 
six o'clock hour circle. 

Let HZO (Fig. 23.) be the meridian of the place, Z the zenith, 
HO the horizon, S the place of „. 23 

the object on the 6 o'clock hour 
circle PSP', which of course cuts 
the equator in the east and west 
points, and ZSB the vertical cir- 
cle passing through the body. 
Then in the right angled triangle 
SBA, the given quantities are 
AS, which is the declination, 
and the arc OP or angle SAB, 
the latitude of the place, to find 
the altitude BS, and the azimuth 
BO, or the amplitude AB, which is its complement. 

Ex. 1. What were the altitude and azimuth of Arcturus, when 




* Degrees are converted into hours by multiplying by 4 and dividing by 60. 



ASTRONOMICAL PROBLEMS. 



65 



upon the six o'clock hour circle of Greenwich, lat. 51° 28' 40" N. 
on the first of April, 1822 ; its declination being 20° 6' 50" N.? 



For the Altitude. 

Rad. sin. BS=sin. AS sin. A 
Rad. . . 10. 
Sin. 20° 06' 50" 9.5364162 
Sin. 51 28 40 9.8934103 



Sin. 15 36 27 



9.4298265 



For the Azimuth. 

Rad. cos. A=cot. BO cot. AS 
Cot. 20° 06' 50" 10.4362545 
Cos. 51 28 40 9.7943612 

Rad. .' . 10. 



Cot. 77° 09' 04' 



9.3581067 




Ex. 2. At latitude 62° 12' N. the altitude of the sun at 6 o'clock 
in the morning was found to be 18° 20' 23": required his declina- 
tion and azimuth. 

Ans. Dec. 20° 50' 12" N. Az. 79° 56' 4". 

135. The Latitudes and Longitudes of two celestial objects be- 
ing given, to find their Distance apart. 

Let P (Fig. 24,) represent the pole of the ecliptic, and PS, PS', 
two arcs of celestial latitude (Art. 37,) drawn to the two objects 
SS' ; then will these arcs represent the Fig. 24 

co-latitudes, the angle P will be the 
difference of longitude, and the arc SS' 
will be the distance sought. Here we 
have the two sides and the included 
angle given to find the third side. By 
Napier's Rules for the solution of oblique angled spherical triangles, 
(see Spherical Trigonometry,) the sum and difference of the two 
angles opposite the given sides may be found, and thence the an- 
gles themselves. The required side may then be found by the theo- 
rem, that the sines of the sides are as the sines of their opposite 
angles.* The computation is omitted here on account of its great 
length. If P be the pole of the equator instead of the ecliptic, 
then PS and PS' will represent arcs of co-declination, and the 
angle P will denote difference of right ascension. From these 
data, also, we may therefore derive the distance between any two 
stars. Or, finally, if P be the pole of the horizon, the angle at P 

* More concise formulae for the solution of this case may be found in Young's Tri- 
gonometry, p. 99. — Francceur's Uranography, Art. 330. — Dr. Bowditch's Practical 
Navigator, p. 436. 

9 



6Q THE EARTH. 

will denote difference of azimuth, and the sides PS, PS', zenith 
distances, from which the side SS' may likewise be determined. 

FIGURE AND DENSITY OF THE EARTH. 

136. We have already shown, (Art. 8,) that the figure of the 
earth is nearly globular ; but since the semi-diameter of the earth 
is taken as the base line in determining the parallax of the heav- 
enly bodies, and lies therefore at the foundation of all astronomi- 
cal measurements, it is very important that it should be ascertained 
with the greatest possible exactness. Having now learned the 
use of astronomical instruments, and the method of measuring 
arcs on the celestial sphere, we are prepared to understand the 
methods employed to determine the exact figure of the earth. 
This element is indeed ascertained in four different ways, each 
of which is independent of all the rest, namely, by investigating 
the effects of the centrifugal force arising from the revolution of 
the earth on its axis — by measuring arcs of the meridian — by 
experiments with the pendulum — and by the unequal action of the 
earth on the moon, arising from the redundance of matter about 
the equatorial regions. We will briefly consider each of these 
methods. 

137. First, the known effects of the centrifugal force, would give 
to the earth a spheroidal figure, elevated in the equatorial, and flat- 
tened in the polar regions. 

Had the earth been originally constituted (as geologists sup- 
pose) of yielding materials, either fluid or semi-fluid, so that 
its particles could obey their mutual attraction, while the body 
remained at rest it would spontaneously assume the figure of a 
perfect sphere ; as soon, however, as it began to revolve on its 
axis, the greater velocity of the equatorial regions would give to 
them a greater centrifugal force, and cause the body to swell out 
into the form of an oblate spheroid.* Even had the solid part of 
the earth consisted of unyielding materials and been created a 
perfect sphere, still the waters that covered it would have receded 
from the polar and have been accumulated in the equatorial re- 

* See a good explanation of this subject in the Edinburgh Encyclopsedia, II. 665. 



FIGURE OF THE EARTH. 67 

gions, leaving bare extensive regions on the one side, and ascend- 
ing to a mountainous elevation on the other. 

On estimating from the known dimensions of the earth and 
the velocity of its rotation, the amount of the centrifugal force in 
different latitudes, and the figure of equilibrium which would 
result, Newton inferred that the earth must have the form of an 
oblate spheroid before the fact had been established by observa- 
tion ; and he assigned nearly the true ratio of the polar and equa- 
torial diameters, f 

1 38. Secondly, the spheroidal figure of the earth is proved, by 
actually measuring the length of a degree on the meridian in differ- 
ent latitudes. 

Were the earth a perfect sphere, the section of it made by a 
plane passing through its center in any direction would be a per- 
fect circle, whose curvature would be equal in all parts ; but if 
we find by actual observation, that the curvature of the section is 
not uniform, we infer a corresponding departure in the earth from 
the figure of a perfect sphere. This task of measuring portions of 
the meridian, has been executed in different countries by means 
of a system of triangles with astonishing accuracy.* The result 
is, that the length of a degree increases as we proceed from the 
equator towards the pole, as may be seen from the following table : 



Places of observation. 


Latitude. 


Length of a degree in mile6. 


Peru, 


00° 00' 00" 


68.732 


Pennsylvania, 

Italy, 

France, 


39 12 00 
43 01 00 
46 12 00 


68.896 
68.998 
69.054 


England, 
Sweden, 


51 29 54£ 
66 20 10 


69.146 
69.292 



Combining the results of various measurements, the dimensions 
of the terrestrial spheroid are found to be as follows : f 

Equatorial diameter, . . . 7925.308 

Polar diameter, .... 7898.952 

Mean diameter, .... 7912.130 



The difference between the greatest and least, is 26.356=3 



* See Day's Trigonometry. t Bessel. 



68 THE EARTH. 

of the greatest. This fraction ( ¥ i T ) is denominated the ellipiicity 
of the earth, being the excess of the transverse over the conjugate 
axis, on the supposition that the section of the earth coinciding 
with the meridian, is an ellipse ; and that such is the case, is 
proved by the fact that calculations on this hypothesis, of the 
lengths of arcs of the meridian in different latitudes, agree nearly 
with the lengths obtained by actual measurement. 

139. Thirdly, the figure of the earth is shown to be spheroidal, hy 
observations with the pendulum. 

The use of the pendulum in determining the figure of the 
earth, is founded upon the principle that the number of vibra- 
tions performed by the same pendulum, when acted on by differ- 
ent forces, varies as the square root of the forces* Hence, by 
carrying a pendulum to different parts of the earth, and counting 
the number of vibrations it performs in a given time, we obtain 
the relative forces of gravity at those places, and this leads to a 
knowledge of the relative distance of each place from the center 
of the earth, and finally, to the ratio between the equatorial and 
the polar diameters. 

140. Fourthly, that the earth is of a spheroidal figure, is infer- 
red from the motions of the moon. 

These are found to be affected by the excess of matter about 
the equatorial regions, producing certain irregularities in the lunar 
motions, the amount of which becomes a measure of the excess 
itself, and hence affords the means of determining the earth's 
ellipticity. This calculation has been made by the most profound 
mathematicians, and the figure deduced from this source corres- 
ponds very nearly to that derived from the several other indepen- 
dent methods. 

We thus have the shape of the earth established upon the most 
satisfactory evidence, and are furnished with a starting point from 
which to determine various measurements among the heavenly 
bodies. 

141. The density of the earth compared with water, that is, its 

* Mechanics, Art. 183. 



DENSITY OF THE EARTH. 



69 




specific gravity, is 5£. # The density was first estimated by Dr. 
Hutton, from observations made by Dr. Maskelyne, Astronomer 
Royal, on Schehallien, a mountain of Scotland, in the year 1774. 
Thus, let M (Fig. 25,) represent Fig. 25. 

the mountain, D, B, two stations 
on opposite sides of the moun- 
tain, and I a star; and let IE 
and IG be the zenith distances as 
determined by the differences of 
latitudes of the two stations. But 
the apparent zenith distances as 
determined by the plumb line 
are IE' and IG'. The deviation 
towards the mountain on each 
side exceeded 7".f The attrac- 
tion of the mountain being ob- 
served on both sides of it, and 
its mass being computed from a number of sections taken in all 
directions, these data, when compared with the known attraction 
and magnitude of the earth, led to a knowledge of its mean den- 
sity. According to Dr. Hutton, this is to that of water as 9 to 2 ; 
but later and more accurate estimates have made the specific 
gravity of the earth as stated above. But this density is nearly 
double the average density of the materials that compose the ex- 
terior crust of the earth, showing a great increase of density 
towards the center. 

The density of the earth is an important element, as we shall 
find that it helps us to a knowledge of the density of each of the 
other members of the solar system. 




* Baily, Ast. Tables, p. 21. 



t Robison's Phys. Ast. 






PART II. OF THE SOLAR SYSTEM. 



142. Having considered the Earth, in its astronomical relations, 
and the Doctrine of the Sphere, we proceed now to a survey of 
the Solar System, and shall treat successively of the Sun, Moon, 
Planets, and Comets. 



CHAPTER I. 

OF THE SUN SOLAR SPOTS ZODIACAL LIGHT. 

143. The figure which the sun presents to us is that of a per- 
fect circle, whereas most of the planets exhibit a disk more or less 
elliptical, indicating that the true shape of the body is an oblate 
spheroid. So great, however, is the distance of the sun, that a 
line 400 miles long would subtend an angle of only I" at the eye, 
and would therefore be the least space that could be measured. 
Hence, were the difference between two conjugate diameters of 
the sun any quantity less than this, we could not determine by 
actual measurement that it existed at all. Still we learn from 
theoretical considerations, founded upon the known effects of cen- 
trifugal force, arising from the sun's revolution on his axis, that 
his figure is not a perfect sphere, but is slightly spheroidal.* 

144. The distance of the sun from the earth, is nearly 95,000,000 
miles. For, its horizontal parallax being 8."6, (Art. 86,) and the 
semi-diameter of the earth 3956 miles, 

Sin. 8."6 : 3956 : : Rad. : 95,000,000 nearly. In order to form 
some faint conception at least of this vast distance, let us reflect 
that a railway car, moving at the rate of 20 miles per hour, would 
require more than 500 years to reach the sun. 

* See Mecanique Celeste, III, 165. Delambre, t. I, p. 483. 



SOLAR SPOTS. 71 

145. The apparent diameter of the sun may be found either by 
the Sextant, (Art. 129,) by an instrument called the Heliometer, 
specially designed for measuring its angular breadth, or by the time 
it occupies in crossing the meridian. If, for example, it occupied 
4 m , its angular diameter would be 1°. It in fact occupies a little 
more than 2 m , and hence its apparent diameter is a little more than 
half a degree, (32' 3"). Having the distance and angular diameter, 
we can easily find its linear diameter. Let E (Fig. 26,) be the 
earth, S the sun, ES a line drawn to the Fig. 26. 
center of the disk, and EC a line drawn 
touching the disk at C. Join SC ; then 

Rad. : ES (95,000,000) : : sin. 16' l."5 : 
442840=semi-diameter, and 885680=diam- 

eter. And— =112 nearly; that is, it 

would require one hundred and twelve bo- 
dies like the earth, if laid side by side, to 
reach across the diameter of the sun ; and a 
ship sailing at the rate of ten knots an hour, 
would require more than ten years to sail 
across the solar disk. Since spheres are to 
each other as the cubes of their diameters, 

l 3 : 112 3 : : 1 : 1,400,000 nearly.; that is, the sun is about 
1,400,000 times as large as the earth. The distance of the moon 
from the earth being 237,000 miles, were the center of the sun 
made to coincide with the center of the earth, the sun would ex- 
tend every way from the earth nearly twice as far as the moon. 

146. In density, the sun is only one fourth that of the earth, 
being but a little heavier than water (Art. 141) ; and since the 
quantity of matter, or mass of a body, is proportioned to its mag- 
nitude and density, hence, 1,400,000 x] = 350,000, that is, the 
quantity of matter in the sun is three hundred and fifty thousand 
(or, more accurately, 354,936) times as great as in the earth. Now 
the weight of bodies (which is a measure of the force of gravity) 
varies directly as the quantity of matter, and inversely as the 
square of the distance. A body, therefore, would weigh 350,000 
times as much on the surface of the sun as on the earth, if the 




72 THE SUN. 

distance of the center of force were the same in both cases ; but 
since the attraction of a sphere is the same as though all the mat- 
ter were collected in the center, consequently, the weight of a 
body, so far as it depends on its distance from the center of force, 
would be the square of 112 times less at the sun than at the earth. 
Or, putting W for the weight at the earth, and W for the weight 
at the sun, then 

w . w .. 1.350000 

Hence a body would weigh nearly 28 times as much at the sun 
as at the earth. A man weighing 200 lbs. would, if transported 
to the surface of the sun, weigh 5,580 lbs., or nearly 2% tons. To 
lift one's limbs, would, in such a case, be beyond the ordinary 
power of the muscles. At the surface of the earth, a body falls 
through 16^2 feet in a second ; and since the spaces are as the 
velocities, the times being equal, and the velocities as the forces, 
therefore a body would fall at the sun in one second, through 
16 T V x 27 T 9 o = 448.7 feet. 

SOLAR SPOTS. 

147. The surface of the sun, when viewed with a telescope, 
usually exhibits dark spots, which vary much, at different times, 
in number, figure, and extent. One hundred or more, assembled 
in several distinct groups, are sometimes visible at once on the 
solar disk. The solar spots are commonly very small, but 
occasionally a spot of enormous size is seen occupying an extent 
of 50,000 miles or more in diameter. They are sometimes 
even visible to the naked eye, when the sun is viewed through 
colored glass, or when near the horizon, it is seen through light 
clouds or vapors. When it is recollected that 1" of the solar 
disk implies an extent of 400 miles, (Art. 143,) it is evident that a 
space large enough to be seen by the naked eye, must cover a very 
large extent. 

A solar spot usually consists of two parts, the nucleus and the 
umbra, (Fig. 27.) The nucleus is black, of a very irregular shape, 
and is subject to great and sudden changes, both in form and size. 
Spots have sometimes seemed to burst asunder, and to project frag- 
ments in different directions. The umbra is a wide margin of lighter 



SOLAR SPOTS. 



73 



shade, and is commonly of greater Fig- 27- 

extent than the nucleus. The spots 
are usually confined to a zone ex- 
tending across the central regions 
of the sun, not exceeding 60° in 
breadth. When the spots are ob- 
served from day to day, they are 
seen to move across the disk of the 
sun, occupying about two weeks in 
passing from one limb to the other. 
After an absence of about the same 
period, the spot returns, having taken 27d. 7h. 37m. in the entire 
revolution. 




148. The spots must be nearly or quite in contact with the body 
of the sun. Were they at any considerable distance from it, the 
time during which they would be seen on the solar disk, would 
be less than that occupied in the remainder of the revolution. 
Thus, let S (Fig. 28,) be the sun, E the earth, and ahc the path 
of the body, revolving about the sun. 
Unless the spot were nearly or quite 
in contact with the body of the sun, 
being projected upon his disk only 
while passing from b to c, and being 
invisible while describing the arc cab, 
it would of course be out of sight lon- 
ger, than in sight, whereas the two pe- 
riods are found to be equal. Moreover, 
the lines which all the solar spots de- 
scribe on the disk of the sun, are found 
to be parallel to each other, like the 
circles of diurnal revolution around the 
earth ; and hence it is inferred that 
they arise from a similar cause, namely, 
the revolution of the sun on his axis, 
a fact which is thus made known to 
us. 

But although the spots occupy about 27| days in passing from 

10 




74 



THE SUN. 




one limb of the sun around to the same limb again, yet this is not 

the period of the sun's revolution on his axis, but exceeds it by 

nearly two days. For, let AA'B (Fig. 29,) represent the sun, and 

EE'M the orbit of the earth. When the earth is at E, the 

visible disk of the sun will be AA'B ; 

and if the earth remained stationary at 

E, the time occupied by a spot after 

leaving A until it returned to A, would 

be just equal to the time of the sun's 

revolution on his axis. But during the 

27J days in which the spot has been 

performing its apparent revolution, the 

earth has been advancing in his orbit 

from E to E', where the visble disk of 

the sun is A'B'. Consequently, before 

the spot can appear again on the limb from which it set out, it 

must describe so much more than an entire revolution as equals 

the arc AA', which equals the arc EE'. Hence, 

365d. 5h. 48m.+27d. 7h. 37m. : 365d. 5h. 48m. : : 27d. 7h. 37m. : 
25d. 9h. 56m.=the time of the sun's revolution on his axis. #- 

149. If the path which the spots appear to describe by the re- 
volution of the sun on his axis left each a visible trace on his sur- 
face, they would form, like the circles of diurnal' re volution on the 
earth, so many parallel rings, of which that which passed through 
the center would constitute the solar equator, while those on each 
side of this great circle would be small circles, corresponding to 
parallels of latitude on the earth. Let us conceive of an artifi- 
cial sphere to represent the sun, having such rings plainly marked 
on its surface. Let this sphere be placed at some distance from 
the eye, with its axis perpendicular to the axis of vision, in which 
case the equator would coincide with the line of vision, and its 
edge be presented to the eye. It would therefore be projected in- 
to a straight line. The same would be the case with all the small- 
er rings, the distance being supposed such that the rays of light 
come from them all to the eye nearly parallel. Now let the axis, 
instead of being perpendicular to the line of vision, be inclined to 
that line, then all the rings being seen obliquely would be projected 



SOLAR SPOTS. 75 

into ellipses. If, however, while the sphere remained in a fixed 
position, the eye were carried around it, (being always in the same 
plane,) twice during the circuit it would be in the plane of the 
equator, and project this and all the smaller circles into straight 
lines ; and twice, at points 90° distant from the foregoing posi- 
tions, the eye would be at a distance from the planes of the rings 
equal to the inclination of the equator of the sphere to the line of 
vision. Here it would project the rings into wider ellipses than 
at other points ; and the ellipses would become more and more 
acute as the eye departed from either of these points, until they 
vanished again into straight lines. 

150. It is in a similar manner that the eye views the paths de- 
scribed by the spots on the sun. If the sun revolved on an axis 
perpendicular to the plane of the earth's orbit, the eye being situ- 
ated in the plane of revolution, and at such a distance frem the 
sun that the light comes to the eye from all parts of the solar 
disk nearly parallel, the paths described by the spots would be 
projected into straight lines, and each would describe a straight 
line across the solar disk, parallel to the plane of revolution. But 
the axis of the sun is inclined to the ecliptic about 7|° from a per- 
pendicular, so that usually^ill the circles described by the spots are 
projected into ellipses. The breadth of these, however, will vary 
as the eye, in the annual revolution, is carried around the sun, and 
when the eye comes into the plane of the rings, as it does twice a 
year, they are projected into straigh^ffies, and for a short time a 
spot seems moving in a straight line inclined to the plane of the 
ecliptic 7^°. The two points where the sun*s equator cuts the 
ecliptic are called the sun's nodes. The longitudes of the nodes 
are 80° 7' and 260° 7', and the earth passes through them about 
the 12th of December, and the 11th of June. It is at these times 
that the spots appear to describe straight lines. We have men- 
tioned the various changes in the apparent paths of the solar spots, 
which arise from the inclination of the sun's axis to the plane of 
the ecliptic ; but it was in fact by first observing these changes, 
and proceeding in the reverse order from that which we have pur- 
sued, that astronomers ascertained that the sun revolves on his 
axis, and that this axis is inclined to the ecliptic 82f°. 



76 



THE SUN. 



151. With regard to the cause of the solar spots, various hypo- 
theses have been proposed, none of which is entirely satisfactory. 
That which ascribes their origin to volcanic action, appears to us 
the most reasonable.* 

Besides the dark spots on the sun, there are also seen, in dif- 
ferent parts, places that are brighter than the neighboring por- 
tions of the disk. These are called faculce. Other inequalities 
are observable in powerful telescopes, all indicating that the sur- 
face of the sun is in a state of constant and powerful agitation. 



ZODIACAL LIGHT. 

152. The Zodiacal Light is a faint light resembling the tail of 
a comet, and is seen at certain seasons of the year following the 
course of the sun after evening twilight, or preceding his approach 
in the morning sky. Figure 30 represents its appearance as seen 
in the evening in March, 1836. The following are the leading 
facts respecting it. 

1 . Its form is that of a luminous Fig. 30. 
pyramid, having its base towards 
the sun. It reaches to an immense 
distance from the sun, sometimes 
even beyond the orbit of the earth. 
It is brighter in the parts nearer the 
sun than in those that are more 
remote, and terminates in an ob- 
tuse apex, its light fading ^vay by 
insensible gradations, until it be- 
comes too feeble for distinct vision. 
Hence its limits are, at the same 
time, fixed at different distances 
from the sun by different observers, 
according to their respective powers 
of vision. 

2. Its aspects vary very much with the different seasons of the 
year. About the first of October, in our climate, (Lat. 41° 18',) 

* In the system of instruction in Yale College, subjects of this kind are discussed 
in a course of astronomical lectures, addressed to the class after they have finished the 
perusal of the text-book. 




ZODIACAL LIGHT. 77 

it becomes visible before the dawn of day, rising along north of 
the ecliptic, and terminating above the nebula of Cancer. About 
the middle of November, its vertex is in the constellation Leo. 
At this time no traces of it are seen in the west after sunset, but 
about the first of December it becomes faintly visible in the west, 
crossing the Milky Way near the horizon, and reaching from the 
sun to the head of Capricornus, forming, as its brightness increases, 
a counterpart to the Milky Way, between which on the right, 
and the Zodiacal Light on the left, lies a triangular space embra- 
cing the Dolphin. Through the month of December, the Zodi- 
acal Light is seen on both sides of the sun, namely, before the 
morning and after the evening twilight, sometimes extending 50° 
westward, and 70° eastward of the sun at the same time. After 
it begins to appear in the western sky, it increases rapidly from 
night to night, both in length and brightness, and withdraws itself 
from the morning sky, where it is scarcely seen after the month 
of December, until the next October. 

3. The Zodiacal Light moves through the heavens in the order of 
the signs. It moves with unequal velocity, being sometimes sta- 
tionary and sometimes retrograde, while at other times it ad- 
vances much faster than the sun. In February and March, it is 
very conspicuous in the west, reaching to the Pleiades and be- 
yond ; but in April it becomes more faint, and nearly or quite dis- 
appears during the month of May. It is scarcely seen in this lat- 
itude during the summer months. 

4. It is remarkably conspicuous at certain periods of a few 
years, and then for a long interval almost disappears. 

5. The Zodiacal Light was formerly held to he the atmosphere of 
the sun.* But La Place has shown that the solar atmosphere 
could never reach so far from the sun as this light is seen to ex- 
tend, j- It has been supposed by others to be a nebulous body 
revolving around the sun. The idea has been suggested, that the 
extraordinary Meteoric Showers, which at different periods visit 
the earth, especially in the month of November, may be derived 
from this body.J 



* Mairan, Memoirs French Academy, for 1733. t Mec. Celeste, III, 525. 

t See note on " Meteoric Showers," at the end of the volume. 



CHAPTER II. 

OF THE APPARENT ANNUAL MOTION OF THE SUN SEASONS FIGURE 

OF THE EARTH'S ORBIT. 

153. The revolution of the earth around the sun once a year, 
produces an apparent motion of the sun around the earth in the 
same period. When bodies are at such a distance from each 
other as the earth and the sun, a spectator on either would pro- 
ject the other body upon the concave sphere of the heavens, al- 
ways seeing it on the opposite side of a great circle, 180° from 
himself. Thus when the earth arrives at Libra (Fig. 11,) we see 
the sun in the opposite sign Aries. When the earth moves from 
Libra to Scorpio, as we are unconscious of our own motion, the 
sun it is that appears to move from Aries to Taurus, being always 
seen in the heavens, where a line drawn from the eye of the spec- 
tator through the body meets the concave sphere of the heavens. 
Hence the line of projection carries the sun forward on one side 
of the ecliptic, at the same rate as the earth moves on the oppo- 
site side ; and therefore, although we are unconscious of our own 
motion, we can read it from day to day in the motions of the sun. 
If we could see the stars at the same time with the sun, we could 
actually observe from day to day the sun's progress through them, 
as we observe the progress of the moon at night ; only the sun's 
rate of motion would be nearly fourteen times slower than that 
of the moon. Although we do not see the stars when the sun is 
present, yet after the sun is set, we can observe that it makes daily 
progress eastward, as is apparent from the constellations of the 
Zodiac occupying, successively, the western sky after sunset, 
proving that either all the stars have a common motion westward 
independent of their diurnal motion, or that the sun has a motion 
past them, from west to east. We shall see hereafter abundant 
evidence to prove, that this change in the relative position of the 
sun and stars, is owing to a change in the apparent place of the 
sun, and not to any change in the stars. 



ANNUAL MOTION. 79 

154. Although the apparent revolution of the sun is in a direc- 
tion opposite to the real motion of the earth, as regards absolute 
space, yet both are nevertheless from west to east, since these 
terms do not refer to any directions in absolute space, but to the 
order in which certain constellations (the constellations of the 
Zodiac) succeed one another. The earth itself, on opposite sides 
of its orbit, does in fact move towards directly opposite points of 
space ; but it is all the while pursuing its course in the order of 
the signs. In the same manner, although the earth turns on its 
axis from west to east, yet any place on the surface of the earth 
is moving in a direction in space exactly opposite to its direction 
twelve hours before. If the sun left a visible trace on the face 
of the sky, the ecliptic would of course be distinctly marked on 
the celestial sphere as it is on an artificial globe ; and were the 
equator delineated in a similar manner, (by any method like that 
supposed in Art. 46,) we should then see at a glance the relative 
position of these two circles, the points where they intersect one 
another constituting the equinoxes, the points where they are at 
the greatest distance asunder, or the solstices, and various other 
particulars, which, for want of such visible traces, we are now 
obliged to search for by indirect and circuitous methods. It will 
even aid the learner to have constantly before his mental vision, 
an imaginary delineation of these two important circles on the 
face of the sky. 

155. The method of ascertaining the nature and position of the 
earth's orbit, is by observations on the sun's Declination and Right 
Ascension. 

The exact declination of the sun at any time is determined 
from his meridian altitude or zenith distance, the latitude of the 
place of observation being known, (Art. 37.) The instant the 
center of the sun is on the meridian, (which instant is given by 
the transit instrument,) we take the distance of his upper and 
that of his lower limb from the zenith : half the sum of the two 
observations corrected for refraction, gives the zenith distance of 
the center. This result is diminished for parallax, (Art. 84,) and 
we obtain the zenith distance as it would be if seen from the 
center of the earth. The zenith distance being known, the de- 



80 THE SUN. 

ciination is readily found, by subtracting that distance from the 
latitude. By thus taking the sun's declination for every day of 
the year at noon, and comparing the results, we learn its motion 
to and from the equator. 

156. To obtain the motion in right ascension, we observe, with 
a transit instrument, the instant when the center of the sun is on 
the meridian. Our sidereal clock gives us the right ascension in 
time (Art. 124,) which we may easily, if we choose, convert into 
degrees and minutes, although it is more common to express right 
ascension by hours, minutes, and seconds. The differences of 
right ascension from day to day throughout the year, give us the 
sun's annual motion parallel to the equator. From the daily re- 
cords of these two motions, at right angles to each other, arran- 
ged in a table,* it is easy to trace out the path of the sun on the 
artificial globe ; or to calculate it with the greatest precision by 
means of spherical triangles, since the declination and right ascen- 
sion constitute two sides of a right angled spherical triangle, the 
corresponding arc of the ecliptic, that is, the longitude, being the 
third side, (Art. 132.) By inspecting a table of observations, 
we shall find that the declination attains its greatest value on 
the 22d of December, when it is 23° 27' 54" south ; that from 
this period it diminishes daily and becomes nothing on the 21st 
of March ; that it then increases towards the north, and reaches 
a similar maximum at the northern tropic about the 22d of June ; 
and, finally, that it returns again to the southern tropic by gra- 
dations similar to those which marked its northward progress. A 
table of observations also would show us, that the daily differences 
of declination are very unequal ; that, for several days, when the 
sun is near either tropic, its declination scarcely varies at all ; 
while near the equator, the variations from day to day are very 
rapid, — a fact which is easily understood, when we reflect, that 
at the solstices the equator and the ecliptic are parallel to each 
other, j- both being at right angles to the meridian ; while at the 

* Such a table may be found in Biot's Astronomy, in Delambre, and in most collec- 
tions of Astronomical Tables. 

t Or, more properly, the tangents of the two circles (which denote the directions of 
the curves at those points) are parallel. 



ANNUAL MOTION. 81 

equinoxes, the ecliptic departs most rapidly from the direction of 
the equator. 

On examining, in like manner, a table of observations of the 
right ascension, we find that the daily differences of right ascen- 
sion are likewise unequal ; that the mean of them all is 3 m 56 s , 
or 236 s , but that they have varied between 215 s and 266 s . On 
examining, moreover, the right ascension at each of the equi- 
noxes, we find that the two records differ by 180°; which proves 
that the path of the sun is a great circle, since no other would 
bisect the equinoctial as this does. %< 

157. The obliquity of the ecliptic is equal to the surfs greatest 
declination. For, by article 22, the inclination of any two great 
circles is equal to their greatest distance asunder, as measured on 
the sphere. The obliquity of the ecliptic may be determined 
from the sun's meridian altitude, or zenith distance, on the day 
of the solstice. The exact instant of the solstice, however, will 
not of course occur when the sun is on the meridian, but may 
happen at some other meridian ; still, the changes of declination 
near the solstice are so exceedingly small, that but a slight error 
can result from this source. The obliquity may also be found, 
without knowing the latitude, by observing the greatest and least 
meridian altitudes of the sun, and taking half the difference. 
This is the method practiced in ancient times by Hipparchus. 
(Art. 2.) On comparing observations made at different periods 
for more than two thousand years, it is found, that the obliquity 
of the ecliptic is not constant, but that it undergoes a slight dimi- 
nution from age to age, amounting to 52" in a century, or about 
half a second annually. We might apprehend that by successive 
approaches to each other the equator and ecliptic would finally 
coincide ; but astronomers have ascertained by an investigation, 
founded on the principles of universal gravitation, that this varia- 
tion is confined within certain narrow limits, and that the obli- 
quity, after diminishing for some thousands of years, will then 
increase for a similar period, and will thus vibrate for ever about 
a mean value. 

158. The dimensions of the eartJCs orbit, when compared with its 
own magnitude, are immense. 

11 



82 THE SUN. 

Since the distance of the earth from the sun is 95,000,000 
miles, and the length of the entire orbit nearly 600,000,000 miles, 
it will be found, on calculation, that the earth moves 1,640,000 
miles per day, 68,000 miles per hour, 1,100 miles per minute, and 
nearly 19 miles every second, a velocity nearly fifty times as great 
as the maximum velocity of a cannon ball. A place on the earth's 
equator turns, in the diurnal revolution, at the rate of about 1,000 
miles an hour and T 5 ¥ of a mile per second. The motion around 
the sun, therefore, is nearly 70 times as swift as the greatest mo- 
tion around the axis. 

THE SEASONS. 

159. The change of seasons depends on two causes, (1) the ob- 
liquity of the ecliptic, and (2) the earth's axis always remaining 
parallel to itself. Had the earth's axis been perpendicular to the 
plane of its orbit, the equator would have coincided with the 
ecliptic, and the sun would have constantly appeared in the equa- 
tor. To the inhabitants of the equatorial regions, the sun would 
always have appeared to move in the prime vertical ; and to the 
inhabitants of either pole, he would always have been in the ho- 
rizon. But the axis being turned out of a perpendicular direc- 
tion 23° 28', the equator is turned the same distance out of the 
ecliptic ; and since the equator and ecliptic are two great circles 
which cut each other in two opposite points, the sun, while per- 
forming his circuit in the ecliptic, must evidently be once a year 
in each of those points, and must depart from the equator of the 
heavens to a distance on either side equal to the inclination of the 
two circles, that is, 23° 28'. (Art. 22.) 

160. The earth being a globe, the sun constantly enlightens 
the half next to him,* while the other half is in darkness. The 
boundary between the enlightened and the unenlightened part, is 
called the circle of illumination. When the earth is at one of 
the equinoxes, the sun is at the other, and the circle of illumina- 

* In fact, the sun enlightens a little more than half the earth, since on account of 
his vast magnitude the tangents drawn from opposite sides of the sun to opposite sides 
of the earth, converge to a point behind the earth, as will be seen by and by in the 
representation of eclipses. The amount of illumination also is increased by refraction. 



THE SEASONS. 



83 



tion passes through both the poles. When the earth reaches one 
of the tropics, the sun being at the other, the circle of illumina- 
tion cuts the earth so as to pass 23° 28' beyond the nearer, and 
the same distance short of the remoter pole. These results would 
not be uniform, were not the earth's axis always to remain parallel 
to itself. The following figure will illustrate the foregoing state- 
ments. 

Fig. 31. 




Let ABCD represent the earth's place in different parts of its 
orbit, having the sun in the center. Let A, C, be the position of 
the earth at the equinoxes, and B, D, its positions at the tropics, 
the axis ns being always parallel to itself.* At A and C the sun 
shines on both n and s ; and now let the globe be turned round 
on its axis, and the learner will easily conceive that the sun will 
appear to describe the equator, which being bisected by the hori- 



* The learner will remark that the hemisphere towards n is above, and that towards 
s is below the plane of the paper. It is important to form a just conception of the 
position of the axis with respect to the plane of its orbit. 



84 THE SUN. 

zon of every place, of course the day and night will be equal in all 
parts of the globe.* Again, at B when the earth is at the south- 
ern tropic, the sun shines 23£° beyond the north pole n, and falls 
the same distance short of the south pole s. The case is exactly 
reversed when the earth is at the northern tropic and the sun at 
the southern. While the earth is at one of the tropics, at B for 
example, let us conceive of it as turning on its axis, and we shall 
readily see that all that part of the earth which lies within the 
north polar circle will enjoy continual day, while that within the 
south polar circle will have continual night, and that all other 
places will have their days longer as they are nearer to the en- 
lightened pole, and shorter as they are nearer to the unenlightened 
pole. This figure likewise shows the successive positions of the 
earth at different periods of the year, with respect to the signs, 
and what months correspond to particular signs. Thus the earth 
enters Libra and the sun Aries on the 21st of March, and on the 
21st of June the earth is just entering Capricorn and the sun Can- 
cer. 

161. Had the axis of the earth been perpendicular to the plane 
of the ecliptic, then the sun would always have appeared to move 
in the equator, the days would every where have been equal to the 
nights, and there could have been no change of seasons. On the 
other hand, had the inclination of the ecliptic to the equator been 
much greater than it is, the vicissitudes of the seasons would have 
been proportionally greater than at present. Suppose, for instance, 
the equator had been at right angles to the ecliptic, in which case, 
the poles of the earth would have been situated in the ecliptic 
itself; then in. different parts of the earth the appearances would 
have been as follows. To a spectator on the equator, the sun as 
he left the vernal equinox would every day perform his diurnal 
revolution in a smaller and smaller circle, until he reached the 
north pole, when he would halt for a moment and then wheel 
about and return to the equator in the reverse order. The pro- 
gress of the sun through the southern signs, to the south pole, 
would be similar to that already described. Such would be the 

* At the pole, the solar disk, at the time of the equinox, appears bisected by the ho- 



FIGURE OF THE EARTHS ORBIT. 85 

appearances to an inhabitant of the equatorial regions. To a 
spectator living in an oblique sphere, in our own latitude for ex- 
ample, the sun while north of the equator would advance continu- 
ally northward, making his diurnal circuits in parallels further and 
further distant from the equator, until he reached the circle of per- 
petual apparition, after which he would climb by a spiral course 
to the north star, and then as rapidly return to the equator. By a 
similar progress southward, the sun would at length pass the circle 
of perpetual occultation, and for some time (which would be 
longer or shorter according to the latitude of the place of obser- 
vation) there would be continual night. 

The great vicissitudes of heat and cold which would attend 
such a motion of the sun, would be wholly incompatible with the 
existence of either the animal or the vegetable kingdoms, and all 
terrestrial nature would be doomed to perpetual sterility and deso- 
lation. The happy provision which' the Creator has made against 
such extreme vicissitudes, by confining the changes of the seasons 
within such narrow bounds, conspires with many other express 
arrangements in the economy of nature to secure the safety and 
comfort of the human race. 



162. Thus far we have taken the earth's orbit as a great circle, 
such being the projection of it on the celestial sphere ; but we now 
proceed to investigate its actual figure. 

Were the earth's path a circle, having the sun in the center, the 
sun would always appear to be at the same distance from us ; that 
is, the radius of its orbit, or radius vector, the name given to a line 
drawn from the center of the sun to the orbit of any planet, 
would always be of the same length. But the earth's distance 
from the sun is constantly varying, which shows that its orbit is 
not a circle. We learn the true figure of the orbit, by ascertain- 
ing the relative distances of the earth from the sun at various pe- 
riods of the year. These all being laid down in a diagram, accord- 
ing to their respective lengths, the extremities, on being connected, 
give us our first idea of the shape of the orbit, which appears of 
an oval form, and at least resembles an ellipse ; and, on further 



86 



THE SUN. 



trial, we find that it has the properties of an ellipse. Thus, let E 
(Fig. 32,) be the place of the earth, and a, b, c, &c. successive po- 
sitions of the sun ; the relative lengths of the lines Ea, Eb, &c. be- 
ing known on connecting the points, a, b, c, &c. the resulting 
figure indicates the true shape of the earth's orbit. 

Fig. 32. 




163. These relative distances are found in two different ways; 
first, by changes in the surfs apparent diameter, and, secondly, by 
variations in his angular velocity. Were the variations in the 
sun's horizontal parallax considerable, as is the case with the 
moon's, this might be made the measure of the relative distances, 
for the parallax varies inversely as the distance, (Art. 82) ; but the 
whole horizontal parallax of the sun is only 9", and its variations 
are too slight and delicate, and too difficult to be found, to serve 
as a criterion of the changes in the sun's distance from the earth. 
But the changes in the surfs apparent diameter, are much more 
sensible, and furnish a better method of measuring the relative 
distances of the earth from the sun. By a principle in optics, the 
apparent diameter of an object, at different distances from the 
spectator, is inversely as the distance.* Hence, the apparent 
diameters of the sun, taken at different periods of the year, be- 
come measures of the different lengths of the radius vector. 



* More exactly, the tangent of the apparent diameter is inversely as the distance ; 
but in small angles like those concerned in the present inquiry, the angle itself may be 
taken for the tangent. 



FIGURE OF THE EARTH'S ORBIT. 87 

164. The point where the earth, or any planet, in its revolution, 
is nearest the sun, is called its perihelion : the point where it is 
furthest from the sun, its aphelion. The place of the earth's peri- 
helion is known, since there the apparent magnitude of the sun is 
greatest ; and when the sun's magnitude is least, the earth is 
known to be at its aphelion. The sun's apparent diameter when 
greatest is 32' 35."6 ; and when least, 31' 31"; hence the radius 
vector at the aphelion : rad. vector at the perihelion : : 32.5933 : 
31.5167 : : 1.034 : 1. Half of the difference of the two is equal 
to the distance of the focus of the ellipse from the center, a quan- 
tity which is always taken as the measure of the 'eccentricity of a* 
planetary orbit. 

165. The differences of angular velocity in the sun in the dif- 
ferent parts of his apparent revolution, are still more remarkable. 
At the perihelion, the sun moves in twenty-four hours over an arc 
of 61', while at the aphelion he describes in the same time an arc 
of only 57', these being the daily increments of longitude in those 
two points respectively. If the apparent motions of the sun de- 
pended alone on our different distances from him, the angular ve- 
locity would vary inversely as the distance, and the ratio expressed 
by these two numbers would be the same as that of the two num- 
bers which denote the differences of apparent diameter in these 

two points. That is, z± (=1.07) would equal ~~ ( = 1.034) ; 

1 57 v ' ^ 31.5167 v ' 

but the first fraction is equal to the square of the second, for 1.07= 
1 .034*. Hence, the sun's angular velocities are to each other inversely 
as the squares of the distances at the perihelion and the aphelion ; and 
by a similar method, the same is found to be true in all points of 
the revolution. 

The angular velocities, therefore, which can be measured very 
accurately by the daily differences of right ascension and declina- 
tion (Art. 132,) converted into corresponding longitudes, enable 
us to determine the different distances of the earth from the sun 
at various points in the orbit. 

166. Since the arcs described by the earth in any small times, 
as in single days, are inversely as the squares of the distances, con- 



88 



THE SUN. 



sequently, the distances are inversely as the square roots of the arcs. 
Upon this principle, the relative distances of the earth from the 
sun, in every point of its revolution, may be easily calculated. 
Thus, we have seen that the arcs described by the sun in one day 
at the perihelion and aphelion are as 61 to 57. Hence the distances 
of the earth from the sun at those two points are as a/57 to >/61, 
or as 1 to 1.034. From twenty- four observations made with the 
greatest care by Dr. Maskelyne at the Royal Observatory of 
Greenwich, the following distances of the earth from the sun are 
determined for each month in the year. 



Time of Observation. 


Distances. 


Time of Observation. 


Distances. 


January 


12-13, 


0.98448 


July 


18-19, 


1.01658 


February 


17-18, 


0.98950 


August 


26-27, 


1.01042 


March 


14-15, 


0.99622 


September 


22-23, 


1.00283 


April 


28-29, 


1.00800 


October 


24-25, 


0.99303 


May 


15-16, 


1.01234 


November 


18-20, 


0.98746 


June 


17-18, 


1.01654 


December 


17-18, 


0.98415 



167. The angular velocity being Fi S- 33 - 
inversely as the square of the distance 
in all parts of the solar orbit, it follows 
that the product of the angle described B; 
in any given time, by the square of the ' a 
distance, is always the same constant 
quantity. For if of two factors, A x 
B, A is increased as B is diminished, 
the product of A and B is always the 
same. If, therefore, from the sun S 
(Fig. 33,) two radii be drawn to T, 
B, the extremities of the arc described in one day, then ST 2 xTB 
gives the same product in all parts of the orbit.* 

168. The radius vector of the solar orbit describes equal spaces 
in equal times, and in unequal times, spaces proportional to the times. 

Let TB (Fig. 33,) be the arc described by the sun in one day ; 
then, Sector TSB-4SB xTB. 

* TB, as seen from the earth, would be projected into a circular arc, equal to the 
measure of the angle at S. 




FIGURE OF THE EARTH'S ORBIT. 89 

Taking Sb as any radius, describe the circular arc ab, which is 
the measure of the angle at S. Now, 

S6 : ab : : SB : BT=SBx^- ; and substituting this value of BT 

in the above equation, we have TSB=iSBxSBx ^=iSB 2 x^, 

So So 

But So is constant, and the product of SB 2 x«o is likewise constant ; 
therefore the sector is always equal to a constant quantity, and 
therefore the radius vector passes over equal spaces in equal 
times.* 

The sun's orbit may be accurately represented by taking some 
point as the perihelion, drawing the radius vector to that point, 
and, considering this line as unity, drawing other radii making 
angles with each other such that the included areas shall be pro- 
portional to the times, and of a length required by the distance of 
each point as given in the table (Art. 166.) On connecting these 
radii, we shall thus see at once how little the earth's orbit departs 
from a perfect circle. Small as the difference appears between 
the greatest and least distances, yet it amounts to nearly -^j of the 
perihelion distance, a quantity no less than 3,000,000 of miles. 

169. The foregoing method of determining the figure of the 
earth's orbit is founded on observation ; but this figure is subject 
to numerous irregularities, the nature of which cannot be clearly 
understood without a knowledge of the leading principles of Uni- 
versal Gravitation. An acquaintance with these will also be in- 
dispensable to our understanding the causes of the numerous ir- 
regularities, which (as will hereafter appear) attend the motions 
of the moon and planets. To the laws of universal gravitation, 
therefore, let us next apply our attention. 

* Francoeur, Uran., p. 62. 

12 



CHAPTER III. 



OF UNIVERSAL GRAVITATION. 



170. Universal Gravitation, is that influence by which every 
body in the universe, whether great or small, tends towards every 
other, with a force which is directly as the quantity of matter, and 
inversely as the square of the distance. 

As this force acts as though bodies were drawn towards each 
other by a mutual attraction, the force is denominated attraction ; 
but it must be borne in mind, that this term is figurative, and im- 
plies nothing respecting the nature of the force. 

The existence of such a force in nature was distinctly asserted 
by several astronomers previous to the time of Sir Isaac Newton, 
but its laws were first promulgated by this wonderful man in his 
Principia, in the year 1687. It is related, that while sitting in a 
garden, and musing on the cause of the falling of an apple, he 
reasoned thus :* that, since bodies far removed from the earth fall 
towards it, as from the tops of towers, and the highest mountains, 
why may not the same influence extend even to the moon ; and 
if so, may not this be the reason why the moon is made to revolve 
around the earth, as would be the case with a cannon ball were 
it projected horizontally near the earth with a certain velocity. 
According to the first law of motion, the moon, if not continually 
drawn or impelled towards the earth by some force, would not 
revolve around it, but would proceed on in a straight line. But 
going around the earth as she does, in an orbit that is nearly cir- 
cular, she must be urged towards the earth by some force, which, 
in a given time, may be represented by the versed sine of the arc 
described in that time. For let the earth (Fig. 34,) be at E, and 
let the arc described by the moon in one second of time be Kb. 
Were the moon influenced by no extraneous force, to turn her 
aside, she would have described, not the arc Ab, but the straight 
line AB, and would have been found at the end of the given time 

* Pemberton's View of Newton's Philosophy. 



UNIVERSAL GRAVITATION. 



91 



at B instead of b. She therefore departs from the line in which 
she tends naturally to move, by the line BZ>, which in small angles 
may be taken as equal to the versed sine A«. This deviation 
from the tangent must be owing to 
some extraneous force. Does this force 
correspond to what the force of gravi- 
ty exerted by the earth, would be at the 
distance of the moon? Now we know the 
distance of the moon from the earth, and 
of course the circumference of her orbit. 
We also know the time of her revolu- 
tion around the earth. Hence we may 
estimate the length of the arc Ab de- 
scribed in one second ; and knowing 
the arc, we can calculate its versed sine. 
For the moon being 60 times as far from the center of the earth, 
as the surface of the earth is from the center, consequently, since 
the force of gravity decreases as the square of the distance in- 
creases,* the space through which the moon would fall by the 




force of the earth's attraction alone, would be 



16 T i 
60 2 



= .05 inches. 



On calculating the value of the versed sine of the arc described in 
one second, it proves to be the same. Hence gravity, and no other 
force than gravity, causes the moon to circulate arouhd the earth. 

171. By this process it was discovered that the law of gravita- 
tion extends to the moon. By subsequent inquiries it was found 
to extend in like manner to all the planets, and to every member 
of the solar system ; and, finally, recent investigations have shown 
that it extends to the fixed stars. The law of gravitation, there- 
fore, is now established as the grand principle which governs all 
the motions of the heavenly bodies. Hence, nothing can be more 
deserving of the attention of the student, than the development of 
the results of this universal law. A few of them only are all that 
can be exhibited in a work like the present : their full develop- 



* Natural Philosophy, Art. 7. That gravity follows the ratio of the inverse square 
of the distance was, however, inferred by Newton from one of Kepler's Laws, to be 
mentioned hereafter. 



92 



UNIVERSAL GRAVITATION. 



ment must be sought for in such great works as the Mecanique 
Celeste of La Place. 

172. If a body revolves about an immovable center of force, and 
is constantly attracted to it, it will always move in the same plane, 
and describe areas about the center proportional to the times. ,* 

Let S (Fig. 35,) be the center of force, and suppose a body to 
be projected at P in the direction of PQR, and take PQ=QR ; 
then, by the first law of motion, the body would move uniformly 
in the direction PQR, and describe PQ, QR, in the same time, if 
no other force acted upon it. But when the body comes to Q, 

Fig. 35. 




let a single impulse act at S, sufficient to draw the body through 
QV, in the time it would have described QR ; and complete the 
parallelogram VQRC, and the body in the same time will describe 
QC ; therefore, PQ, QC, are described in the same time. But 
the triangle SCQ=SRQ=SPQ ; that is, equal areas are described 
in equal times. For the same reason, if a single impulse act at 
C, D, E, &c. at equal intervals of time, the several areas SPQ, 
SQC, SCD, SDE, &c. will all be equal to each other. Now this 

* The learner will remark that what has been before proved (Art. 168,) respecting 
the radius vector of the earth, is here shown to hold good with respect to every body 
which revolves around a center of force ; and the same is true of several other propo- 
sitions demonstrated in this chapter. 



UNIVERSAL GRAVITATION. 



93 



demonstration is independent of any particular dimensions in the 
several triangles, and consequently holds good when they are 
taken indefinitely small, in which case we may consider the force 
as acting, not by separate impulses, but constantly, causing the 
body to describe a curve around S. And as no force acts out of 
the plane SPQ, the whole curve must lie in that plane ; that is, 
the body moves always in the same plane. 

1 73. If a body describes a curve around a center towards which it 
tends by any force, the angular velocity of the body around that center 
is reciprocally as the square of the distance from it.* 

Let ABE (Fig. 36,) be any curve de- Fig. 36. 

scribed about the center S ; draw SA, SB, 
to any two points of the curve A and B ; 
and let AD, BE, be described in indefi- 
nitely small equal times. Join SD and 
SE, and with the center S and distance 
SD, describe a circle meeting SA, SB, SE, 
in F, G, H ; and with the center S and 
distance SE describe a circle meeting SB 
inK. 

Because AD and BE are described in 
equal times, the triangles ASD, BSE, are 
equal. Hence, (Euc. 15. 6.) 
DF : EK :: BS : ASf :: BS 2 : BSxAS (1) 
GH : EK :: SH : SE :: SF : SE :: SA : SB :: SA 2 : BSxAS (2) 

Hence, (1) DF : BS 2 : : EK : BS x AS 

(2) GH: AS 2 :: EK: BSxAS 
.-. DF:GH::BS 2 : AS 2 . 

But DF and GH measure the respective angular velocities at 
A and B, while AS and BS represent the distance at the same 
points. Therefore the angular velocities are reciprocally as the 
squares of the distances. J 







174. In the same curve, the velocity, at any point of the curve, 

* It will be remarked that this is a general proposition, of which article 165 affords 
a particular example. 

t DF and EF are considered as the altitudes of the triangles respectively. 
t Stewart's Phys. and Math. Essays. 



94 UNIVERSAL GRAVITATION. 

varies inversely as the perpendicular drawn from the center of 

force to the tangent at that point. 

Draw SY (Fig. 35,) perpendicular to QP produced ; then the 

area SPQ=iPQ x SY, which varies as PQ x SY /. PQ x 

area SPQ area SPQ AT 
gY But PQ a V, the velocity at P. \ Vac- ^-^- Now 

in the curve described from P, with a constant force, SY becomes 
a perpendicular to the tangent to the curve. But by article 
172, the area described in a given time is constant. Therefore 

SPQ is constant, and V a ; that is, the velocity varies inverse- 

SY 

ly as the perpendicular upon the tangent. Hence, the velocity of 
a revolving body increases as it approaches the center of force. 

175. If equal areas be described about a center in equal times, 
the force must tend towards that center. 

Let SPQ (Fig. 35,)=SQC ; now SPQ=SQR /. SQC-SQR.-. 
CR is parallel to QS. Complete the parallelogram QRCV, and 
by the supposition the body describes QC, in consequence of the 
impulse at Q, and it would have described QR if no such impulse 
had acted ; therefore QV must represent that motion impressed 
at Q, which, in conjunction with the motion QR, can make a body 
describe QC, and QV is directed to S. 

176. Now it appears from article 168, that it is a fact, derived 
from observation, that the earth's radius vector describes equal 
areas in equal times ; and by similar observations the same is 
found to be true of each of the primary planets about the sun, 
and of each of the satellites about its primary. Hence, it is in- 
ferred, that the primary planets all gravitate towards the sun, and 
that the secondary planets all gravitate towards their respective 
primaries. 

It has further been established by observation, (Art. 162,) that 
the planetary orbits are ellipses ; and hence the application of the 
principles of gravitation, so far as respects the sun and planets, 
may be confined to the consideration of the motion of a body in 
an elliptical orbit. 

111. The distance of any planet from the sun at any point in its 

■ 



UNIVERSAL GRAVITATION. 



95 



orbit, is to its distance from the superior focus, as the square of its 
velocity at its mean distance from the sun, is to the square of its ve- 
locity at the given point. 

Let ADBE (Fig. 37,) be the orbit of a planet, S the focus in 
which the sun is placed, AB the transverse and DE the conjugate 
axis, C the center, and F the superior focus. Let the planet be 
any where at P ; and draw a tangent to the orbit at P, on which 
from the foci let fall the perpendiculars SG, FH. Draw also DK 
touching the orbit in D, and let SK be perpendicular to it. Let 

Fig. 37. 

N 




the velocity of the planet when at the mean distance at D=C, and 
when at P=V. Join SP, FP. Then (Art. 174,) the velocity at 
D is to the velocity at P, as SG to SK ; that is, 
C: V::SG:DC. 
C 2 : V 2 : : SG 2 : DC 2 . 
But because the triangles SGP, FHP, are equiangular, having 
right angles at G and H, and also, from the nature of the ellipse, 
the angles SPG, FPH, equal, 

SP : PF : : SG : FH : : SG 2 : CD 2 =FHxSG 
.-. SP : PF : : C 2 : V 2 

178. If of two bodies gravitating to the same center, one descends 
in a straight line, and the other revolves in a curve ; then, if the ve- 
locities of these bodies are equal in any one case, when they are 



96 



UNIVERSAL GRAVITATON. 



Fig. 38. 




equally distant from the center, they will always be equal when they 
are equally distant from it. 

Let ABC (Fig. 38,) be a curve which a body 
describes about a center S to which it gravi- 
tates, while another body descends in a 
straight line AS to that center. Let BC be 
any arc of the curve ABC, and let BD, CH, 
be arcs of circles described from the center 
S, intersecting the line AS in D and H. 
From the center S describe the arc bd, in- 
definitely near to BD, and draw Ef perpen- 
dicular to Bb. Then, because the distances 
SD and SB are equal, the forces of gravity 
at D and B are also equal. Let these forces 
be expressed by the equal lines Dd and BE ; 
and let the force BE be resolved into the 
forces Ef and Bf The force Ef acting at 
right angles to the path of the body, will not affect its velocity in 
that path, but will only draw it aside from a rectilinear course and 
make it proceed in the curve BbC. But the other force Bf acting 
in the direction of the course of the body, will be wholly employed 
in accelerating it. And because B and b are indefinitely near to 
each other, and likewise D and d, the accelerating force from B to 
b and from D to d, may be considered as acting uniformly. 
Therefore, the accelerations of the bodies in D and B, produced 
in equal times, are as the lines Dd, Bf; and hence, putting d for 
the increment of velocity at d, and /for the increment of velocity 
at/, 

d :f::Vd or BE :Bf(l) 

And because the angle at E is a right angle, 

BE*=BbxBf..BE=VBbxVBf..BEx^Bf=VBbxBf. 

Hence, BE : Bf: : VBb : VBf (2) 

And, (1) and (2), d :/: : VBb : VBf (3) 

But, putting b for the velocity at b, and observing that, in falling 
bodies, the velocities are as the square roots of the spaces, 
b:f::VBb:VBf(4) 

Therefore, (3) and (4), b:f::d :/.-. b—d ; that is, the velocity at 
b equals the velocity at d. And, since the same reasoning holds 



ve- 



UNIVERSAL GRAVITATION. 97 

for successive points that may be taken at equal distances from B 
and D, therefore, if of two bodies, &c* 

179. The law according to which the planets gravitate is such, that 
any body under the influence of the same force, and falling direct to 
the sun, will have its velocity at any point equal to a constant velocity 
multiplied into the square root of the distance it has fallen through, 
divided by the square root of the distance between the body and the 
sun's center. 

Suppose a planet to revolve in the elliptical orbit APB (Fig. 37); 

(AFY 1 
— - r (Art. 177) ; or 

(ANX 1 - 
-— r. Let a body at 

A begin to descend towards S with this velocity, then if SL=SP, 
the velocity of the planet at P will be the same as that of the fall- 
ing body at L, (Art. 178.) But the velocity of the planet at P is 

— y =C ( -— 1 ' . But this velocity is equal to the constant 

locity expressed by C, multiplied into the square root of NL, the 
distance fallen through,! divided by the square root of LS, the 
distance between the body and the sun's center.§ 

180. The force with which any planet gravitates to the sun, is in- 
versely as the square of its distance from the sun's center. 

Let C (Fig. 39,) be the center to which the falling body gravi- 
tates, A the point from which it begins to fall, and its velocity at 
any point B, is to its velocity in the point G, which bisects AC, as 

(A T>\ 1 
=— = , r : l.|| Let DEF be a curve such that if AD be an ordinate 

or a perpendicular to AC, meeting the curve in D, and BE any other 

* Principia, Lib. i, Pr. 40. Stewart's Math, and Pays. Essays, Pr. 13. 
t For SN=AB=SP+PF=:SP+NL .-. PF=NL. 

X That NL (=PF) is the distance fallen through to acquire the velocity at P, is de- 
monstrated by writers on Central Forces. (See Vince, Syst. Ast., Art. 823.) 
§ Playfair, Phys. Ast. 
|| For, denoting the velocity at B by V, and the velocity at G by V, 

. , SS)*"- 

13 



^^^MfHffiMi 



98 



UNIVERSAL GRAVITATION. 



ordinate, AD is to BE as the force at A to the force at B, then 



will twice the area ABED be equal to the 
square of the velocity which the body has 
acquired in B. # If therefore the velocity ^ 
at B be V, that at the middle point G being c, * 

Y=c (!?)* and therefore 2ABED=c 2 . ^?; 

and since AB = AC - BC, 2ABED = c 2 . g 

AC-BC 9 /AC \ _, & 
— jtt^ — — c Itt; ^r~ lj- r or the same reason, 
BO \BC / 

if be be drawn indefinitely near to BE, 2A6eD 



Fig. 39. 



/ AC 

Uc 



1 ), and therefore the difference of 



these areas, or 2B6eE, that is, 2EBxB6=c 2 
AC AC\_ 3 AC(BC-&C)_ 2 AC xBb 



CI 




/AC AC\ 
\ bC BC/ 



BCx6C 



AC 

B5,2EB=c 2 .^; orEB 



AG 



, . Wherefore, dividing by 
JdC ' 

now c 2 and AG are constant 



BC 2 ' 

quantities, therefore EB varies inversely as BC 2 . But EB repre- 
sents the force of gravity at B, and BC the distance from the 
sun. Therefore, the force of gravity of a planet in different parts 
of its orbit, is inversely as the square of its distance from the sun. 



181. The line CG is the same with the mean distance of the 
planet in an orbit of which AC is the length of the transverse axis ; 
and if the gravitation at that distance=F, and the mean distance 

itself =«, then since EB=c 2 9 ? F=c 



BC 



x - =— , or <zF=c 2 . 
a* a 



* This principle is demonstrated by the aid of Fluxions as follows : 

By construction, BE is proportional to the force at B= -j-, v being the velocity 

which the moving body has acquired at B, and t the time of the descent from A to B. 
Now B6 is the momentary increment of B A the space, and therefore=?)<ft ; therefore 
BExBb—vdv. And 2BExBb=2vdv. But BE xB& is the momentary increment of 
the area ABED, and 2vdv is the momentary increment of v 2 ; therefore the square of 
the velocity of the moving body, and twice the area of ABED, increase at the same 
rate, and begin to exist at the same time ; therefore they are equal. (See Playfair's 
Outlines, Mechanics, Art. 96.) 

t bC being ultimately equal to BC. 



UNIVERSAL GRAVITATION. ' 99 

182. The squares of the times of revolution of any two planets, 
are as the cubes of their mean distances from the sun. 

If a be the mean distance, or the semi-transverse axis, h the 
semi-conjugate, then tf#6=area of the orbit.* But as c is the ve- 
locity at the mean distance, or the elliptic arch which the planet 
moves over in a second when it is at D, (Fig. 37.) the vertex of the 
conjugate axis, therefore \bc is the area described in that second 
by the radius vector ; and since the area is the same for every 
second of the planet's revolution (Art. 172,) therefore the area of 
the orbit divided by \bc will give the number of seconds in 

which the revolution is completed, which=^-r- =— ; or, since 

\bc c 

c 2 = a¥, (Art. 181,) the time of a revolution = — ==2* V/-fr. 

VdF v * 

Hence, let t, t', be the times of revolutions for two different plan- 
ets, of which the mean distances are a, d , and the force of gravity 

at those distances F, P. Then t : t' : : 2* y fr : %* S/ xv : : \/p : 
J^\ £:**::£*£. But (Art. 180,) F : P : : a' 3 : a* .\ f : t>°~ : :%: 
: a 3 : a' 3 . That is, the squares of the times are as the 



n/' 



~, or Z 2 : " s 

cubes of the mean distances ; or, since the major axes of the or- 
bits are double the mean distances, the squares of the times are as 
the cubes of the major axes. 

183. This is one of Kepler's three great Laws, which, taken in 
connexion, are as follows : 

1. The orbits of all the planets are ellipses, the sun occupying the 
common focus, (Art. 176.) 

2. The radius vector of any planet describes areas proportional 
to the times, (Art. 172.) 

3. The squares of the periodical times are as the cubes of the ma- 
jor axes of the orbits. (Art. 182.) 

These great and fundamental principles of the planetary mo- 
tions, were discovered by the illustrious Kepler by long and as- 
siduous study of the observations made by Tycho Brahe, and 

* Day's Mensuration. 



100 UNIVERSAL GRAVITATION. 

hence he has been called the legislator of the skies. They, there- 
fore, became known as facts, before they were demonstrated 
mathematically. The glory of this achievement was reserved 
for Newton, who proved that they were necessary results of the 
law of universal gravitation. 



MOTION IN AN ELLIPTICAL ORBIT. 

184. Having now acquired some knowledge of the law of uni- 
versal gravitation, let us next endeavor to gain a just conception 
of the forces by which the planets are made to revolve in their 
orbits about the sun. In obedience to the first law of motion, 
every moving body tends to move in a straight line ; and were not 
the planets deflected continually towards the sun by the force of 
attraction, these bodies as well as others would move forward in 
a rectilineal direction. We call the force by which they tend to 
such a direction the projectile force, because its effects are the 
same as though the body were originally projected from a certain 
point in a certain direction. It is an interesting problem for me- 
chanics to solve, what was the nature of the impulse originally 
given to the earth, in order to impress upon it its two motions, the 
one around its own axis, the other around the sun ? If struck in 
the direction of its center of gravity it might receive a forward 
motion, but no rotation on its axis. It must, therefore, have been 
impelled by a force, whose direction did not pass through its cen- 
ter of gravity. Bernouilli, a celebrated mathematician, has calcu- 
lated that the impulse must have been given very nearly in the 
direction of the center, the point of projection being only the 165th 
part of the earth's radius from the center.* This impulse alone 
would cause the earth to move in a right line : gravitation towards 
the sun causes it to describe an orbit. Thus a top spinning on a 
smooth plane, as that of glass or ice, if impelled in a direc- 
tion not passing through the center of gravity, may be made to 
imitate the two motions of the earth, especially if the experiment 
is tried in a concave surface like that of a large bowl. The re- 
sistance occasioned by the surface on which the top moves, and 

* Francoeur, Uran. p. 49. 



UNIVERSAL GRAVITATION. 



101 



that of the air, will generally destroy the force of projection and 
cause the top to revolve in a smaller and smaller orbit ; but the 
earth meets with no such resistance, and therefore makes both her 
days and years of the same length from age to age. A body, 
therefore, revolving in an orbit about a center of attraction, is 
constantly under the influence of two forces, — the projectile force, 
which tends to carry it forward in a straight line which is a tan- 
gent to its orbit, and the centripetal force, by which it tends to- 
wards the center. 



185. The most simple example we have of the combined action 
of these two forces is the motion of a missile thrown from the 
hand, or of a ball fired from a cannon. It is well known that the 
particular form of the curve described by the projectile, in either 
case, will depend upon the velocity with which it is thrown. In 
each case the body will begin to move in the line of direction in 
which it is projected, but it will soon be deflected from that line 
towards the earth. It will however continue nearer to the line of 
projection as the velocity of projection is greater. Thus let AB 

Fig. 40. 
A B 




(Fig. 40,) perpendicular to AC represent the line of projection. 
The body will, in every case, commence its motion in the line AB, 
which will therefore be the tangent to the curve it describes ; but 
if it be thrown with a small velocity, it will soon depart from the 
tangent, describing the line AD ; with a greater velocity it will 
describe a curve nearer to the tangent, as AE ; and with a still 
greater velocity it will describe the curve AF. 

As an example of a body revolving in an orbit under the influ- 
ence of two forces, suppose a body placed at any point P (Fig. 40') 
above the surface of the earth, and let PA be the direction of the 
earth's center ; that is, a line perpendicular to the horizon. If the 



102 



UNIVERSAL GRAVITATION, 




body were allowed to move without receiving any impulse, it 
would descend to the earth in the direction PA with an accelerated 
motion. But suppose that, at the moment of its departure from 
P, it receives a blow in the direction PB, which would carry it to 
B in the time the body would fall from P to A ; then, under the in- 
fluence of both forces, it would descend along the curve PD. If 
a stronger blow were given to it in the direction PB, it would de- 
scribe a larger curve, PE ; or, finally, if the impulse were suffi- 
ciently strong, it would circulate quite around the earth, and re- 
turn again to P, describing the circle PFG. With a velocity of 
projection still greater, it would describe an ellipse, PIK ; and if 
the velocity were increased to a certain degree, the figure would 
become a parabola or hyperbola LMP, and never return into 
itself. 



186. In figure 41, suppose the planet to have passed the point C 
with so small a velocity, that the attraction of the sun bends its 
path very much, and causes it immediately to begin to approach 
towards the sun ; the sun's attraction will increase its velocity as 
it moves through D, E, and F. For the sun's attractive force on 
the planet, when at D, is acting in the direction DS, and, on account 
of the small inclination of DE to DS, the force acting in the line 
DS helps the planet forward in the path DE, and thus increases 
its velocity. In like manner the velocity of the planet will be con- 
tinually increasing as it passes through D, E, and F ; and though 
the attractive force, on account of the planet's nearness, is so much 
increased, and tends, therefore, to make the orbit more curved, 



UNIVERSAL GRAVITATION. 



103 




yet the velocity is also so much increased, that the orbit is not 
more curved than before. The same increase of velocity occa- 
sioned by the planet's approach to the sun, produces a greater in- 
crease of centrifugal force which carries it off again. We may 
see also why, when the planet has Fig. 41. 

reached the most distant parts of its 
orbit, it does not entirely fly off, and 
never return to the sun. For when 
the planet passes along H, K, A, the 
sun's attraction retards the planet, 
just as gravity retards a ball rolled up 
hill ; and when it has reached C, its 
velocity is very small, and the attrac- 
tion at the center of force causes a 
a great deflection from the tangent, 
sufficient to give its orbit a great cur- 
vature, and the planet turns about, returns to the sun, and goes 
over the same orbit again.* As the planet recedes from the sun, 
its centrifugal force diminishes faster than the force of gravity, so 
that the latter finally preponderates-! 

187. We may imitate the motion of a body in its orbit by sus- 
pending a small ball from the ceiling by a long string. The ball 
being drawn out of its place of rest, (which is directly under the 
point of suspension,) it will tend constantly towards the same 
place by a force which corresponds to the force of attraction of a 
central body. If an assistant stands under the point of suspen- 
sion, his head occupying the place of the ball when at rest, the 
ball may be made to revolve about his head as the earth or any 
planet revolves about the sun. By projecting the ball in different 
directions, and with different degrees of velocity, we may make 
it describe different orbits, exemplifying principles which have 
been explained in the foregoing propositions. 



* Airy. 

t The centrifugal force varies inversely as the cube of the distance, while the force 
of gravity is inversely as the square. The centrifugal force, therefore, increases faster 
than the force of gravity as a body is approaching the sun, and decreases faster as the 
body recedes from the sun. (See M. Stewart's Phys. and Math. Tracts, Prop. 8.) 






607 



CHAPTER IV 



PRECESSION OF THE EQUINOXES NUTATION ABERRATION MOTION 

OF THE APSIDES MEAN AND TRUE PLACES OF THE SUN. 

188. The Precession of the Equinoxes, is a slow but continual 
shifting of the equinoctial points from east to west. 

Suppose that we mark the exact place in the heavens, where, 
during the present year, the sun crosses the equator, and that this 
point is close to a certain star ; next year the sun will cross the 
equator a little way westward of that star, and so every year a 
little further westward, until, in a long course of ages, the place 
of the equinox will occupy successively every part of the ecliptic, 
until we come round to the same star again. As, therefore, the 
sun, revolving from west to east in his apparent orbit, comes 
round towards the point where it left the equinox, it meets the 
equinox before it reaches that point. The appearance is as though 
the equinox goes forward to meet the sun, and hence the phenom- 
enon is called the Precession of the Equinoxes, and the fact is 
expressed by saying that the equinoxes retrograde on the ecliptic, 
until the line of the equinoxes makes a complete revolution from 
east to west. The equator is conceived as sliding westward on 
the ecliptic, always preserving the same inclination to it, as a ring 
placed at a small angle with another of nearly the same size, 
which remains fixed, may be slid quite around it, giving a cor- 
responding motion to the two points of intersection. It must be 
observed, however, that this mode of conceiving of the precession 
of the equinoxes is purely imaginary, and is employed merely for 
the convenience of representation. 

189. The amount of precession annually is 50." 1 ; whence, 
since there are 3600" in a degree, and 360° in the whole circum- 
ference, and consequently, 1296000", this sum divided by 50.1 
gives 25868 years for the period of a complete revolution of the 
equinoxes. 



PRECESSION OF THE EQUINOXES. 105 

190. Suppose now we fix to the center of each of the two 
rings (Art. 188) a wire representing its axis, one corresponding to 
the axis of the ecliptic, the other to that of the equator, the ex- 
tremity of each being the pole of its circle. As the ring deno- 
ting the equator turns round on the ecliptic, which with its axis 
remains fixed, it is easy to conceive that the axis of the equator 
revolves around that of the ecliptic, and the pole of the equator 
around the pole of the ecliptic, and constantly at a distance equal 
to the inclination of the two circles. To transfer our conceptions 
to the celestial sphere, we may easily see that the axis of the diur- 
nal sphere, (that of the earth produced, Art. 28,) would not have 
its pole constantly in the same place among the stars, but that this 
pole would perform a slow revolution around the pole of the 
ecliptic from east to west, completing the circuit in about 26,000 
years. Hence the star which we now call the pole star, has not 
always enjoyed that distinction, nor will it always enjoy it here- 
after. When the earliest catalogues of the stars were made, this 
star was 12° from the pole. It is now 1° 24', and will approach 
still nearer ; or, to speak more accurately, the pole will come still 
nearer to this star, after which it will leave it, and successively 
pass by others. In about 13,000 years, the bright star Lyra, 
which lies on the circle of revolution opposite to the present pole 
star, will be within 5° of the pole, and will constitute the Pole 
Star. As Lyra now passes near our zenith, the learner might 
suppose that the change of position of the pole among the stars, 
would be attended with a change of altitude of the north pole 
above the horizon. This mistaken idea is one of the many mis- 
apprehensions which result from the habit of considering the 
horizon as a fixed circle in space. However the pole might shift 
its position in space, we should still be at the same distance from 
it, and our horizon would always reach the same distance be- 
yond it. 

191. The precession of the equinoxes is an effect of the spheroidal 
figure of the earth, and arises from the attraction of the sun and 
moon upon the excess of matter about the earth's equator. 

Were the earth a perfect sphere the attractions of the sun and 
moon upon the earth would be in equilibrium among themselves. 

14 



106 THE SUN. 

But if a globe were cut out of the earth, (taking half the polar 
diameter for radius,) it would leave a protuberant mass of matter 
in the equatorial regions, which may be considered as all collected 
into a ring resting on the earth. The sun being in the ecliptic, 
while the plane of this ring is inclined to the ecliptic 23° 28', of 
course the action of the sun is oblique to the ring, and may be 
resolved into two forces, one in the plane of the equator, and the 
other perpendicular to it. The latter only can act as a disturbing 
force, and tending as it does to draw down the ring to the ecliptic, 
the ring would turn upon the line of the equinoxes as upon a 
hinge, and dragging the earth along with it, the equator would 
ultimately coincide with the ecliptic were it not for the revolution 
of the earth upon its axis. This may be better understood by the 
aid of a diagram. Let TAB (Fig. 42,) represent the equator, 

Fig. 42. 

E 



cf> f 

TED the ecliptic, and AD the solstitial colure. Let AB be the 
movement of rotation for a very short time, being of course in the 
order of the signs and in the direction of the equator. Let BC be 
the movement produced by the disturbing force of the sun in the 
same time. The point A will describe the diagonal AC, the equa- 
tor will take the inclined situation CAT' ; the equinoctial point 
will retrograde from T to T' ; the colure AD will take the posi- 
tion AE, while the inclination of the two planes, that is, the ob- 
liquity of the ecliptic, will remain nearly the same.* 

192. The moon conspires with the sun in producing the pre- 
cession of the equinoxes, its effect, on account of its nearness to 
the earth, being more than double that of the sun, or as 7 to 3. 
The planets likewise, by their attraction, produce a small effect 

* Delambre, t. 3, p. 145. Playfair's Outlines, 2, 308. 



PRECESSION OP THE EQUINOXES. 107 

upon the equatorial ring, but the result is slightly to diminish the 
amount of precession. The whole effect of the sun and moon 
being 50."41, that of the planets is 0.31, leaving the actual amount 
of precession 50." 1.* 

This effect is not to be imagined as taking place merely at the 
time of the equinoxes, but as resulting constantly from the action 
of the sun and moon on the equatorial ring, and at every revolu- 
tion of this ring along with the earth on its axis. Conceive of 
any point in the ring, and follow it round in the diurnal revolution, 
and it will be seen that that point, in consequence of the attrac- 
tion of the sun and moon, will be made to cross the ecliptic a little 
further westward than on the preceding day. 

193. The time occupied by the sun in passing from the equinoc- 
tial point round to the same point again, is called the tropical year. 
As the sun does not perform a complete revolution in this inter- 
val, but falls short of it 50." 1, the tropical year is shorter than the 
sidereal by 20m. 20s. in mean solar time, this being the time of 
describing an arc of 50." 1 in the annual revolution.-)- The 
changes produced by the precession of the equinoxes in the ap- 
parent places of the circumpolar stars, have led to some interest- 
ing results in chronology. In consequence of the retrograde mo- 
tion of the equinoctial points, the signs of the ecliptic (Art. 35,) 
do not correspond at present to the constellations which bear the 
same names, but lie about one whole sign or 30° westward of 
them. Thus, that division of the ecliptic which is called the sign 
Taurus, lies in the constellation Aries, and the sign Gemini in the 
constellation Taurus. Undoubtedly, however, when the ecliptic 
was thus first divided, and the divisions named, the several con- 
stellations lay in the respective divisions which bear their names. 
How long is it, then, since our zodiac was formed ? 

50."1 : 1 year :: 30°(=108000") : 2155.6 years. 

The result indicates that the present divisions of the zodiac 
were made soon after the establishment of the Alexandrian school 
of astronomy. (Art 2.) 

* Francoeur, Uran. 162. t 59' 8."3 : 24h. : : 50."1 : 20m. 20s. 



108 THE SUN. 



NUTATION. 



194. Nutation is a vibratory motion of the earth's axis, arising 
from periodical fluctuations in the obliquity of the ecliptic. 

If the sun and moon moved in the plane of the equator, there 
would be no precession, and the effect of their action in producing 
it varies with their distance from that plane. Twice a year, there- 
fore, namely, at the equinoxes, the effect of the sun is nothing ; 
while at the solstices the effect of the sun is a maximum. On 
this account, the obliquity of the ecliptic is subject to a semi-an- 
nual variation, since the sun's force which tends to produce a 
change in the obliquity is variable, while the diurnal motion of 
the earth which prevents the change from taking place, is con- 
stant. Hence the plane of the equator is subject to an irregular 
motion which is called the Solar Nutation. The name is derived 
from the oscillatory motion communicated by it to the earth's axis, 
while the pole of the equator is performing its revolution around 
the pole of the ecliptic (Art. 190.) The effect of the sun however 
is less than that of the moon, in the ratio of 2 to 5. By the nuta- 
tion alone the pole of the earth would perform a revolution in a 
very small ellipse, only 18" in diameter, the center being in the 
circle which the pole describes around the pole of the ecliptic ; 
but the combined effects of precession and nutation convert the 
circumference of this circle into a wavy line. The motion of the 
equator occasioned by nutation, causes it alternately to approach 
to and recede from the stars, and thus to change their declinations. 
The solar nutation, depending on the position of the sun with re- 
spect to the equinoxes, passes through all its variations annually ; 
but the lunar nutation depending on the position of the moon with 
respect to her nodes, varies s through a period of about 18£ years. 

ABERRATION. 

195. Aberration is an apparent change of place in the stars, 
occasioned by the joint effects of the motion of the earth in its orbit, 
and the progressive motion of light. 

Let EE' (Fig. 43,) represent a part of the earth's orbit, and SE 
a ray of light from the star S. Take EC and EA proportional 



MOTION OF THE APSIDES. 



109 




to the velocity of each respectively ; com- 
plete the parallelogram, and draw the diagonal 
EB. Since an object always appears in the 
direction in which a ray of light coming from 
it, meets the eye, the combination of the two 
motions produces an impression on the eye 
exactly similar to that which would have been 
produced if the eye had remained at rest in 
the point E, and the particle of light had come 
down to it in the direction S'E ; the star, 
therefore, whose place is at S, will appear to 
the spectator at E to be situated at S'. The 
difference between its true and its apparent place, that is, the 
angle SES' is the aberration, the magnitude of which is obtained 
from the known ratio of EA to EC, or the velocity of light to that 
of the earth in its orbit. 

The velocity of light is 192,000 miles per second, while that of 
the earth in its orbit is about 19 miles per second. Represent- 
ing the velocity of light by the line EA, and that of the earth by 
AB, then, 

192,000 : 19: : Rad. : tan. 20."5=the angle at E, which is the 
amount of aberration when the direction of the ray of light is per- 
pendicular to the earth's motion. 

The effect of aberration upon the places of the fixed stars is to 
carry their apparent places a little forward of their real places in 
the direction of the earth's motion. The effect upon each particu- 
lar star will be to make it describe a small ellipse in the heavens, 
having for its center the point in which the star would be seen if 
the earth were at rest. 



MOTION OF THE APSIDES. 



196. The two points of the ecliptic where the earth is at the 
greatest and least distances from the sun respectively, do not 
always maintain the same places among the signs, but gradually 
shift their positions from west to east. If we accurately observe 
the place among the stars, where the earth is at the time of its 
perihelion the present year, we shall find that it will not be pre- 



llU THE SUN. 

cisely at that point the next year when it arrives at its perihelion, 
but about 12" (11. "66) to the east of it. And since the equinox 
itself, from which longitude is reckoned, moves in the opposite 
direction 50."1 annually, the longitude of the perihelion increases 
every year 61. "76, or a little more than one minute. This fact 
is expressed by saying that the line of the apsides of the earth's 
orbit has a slow motion from west to east. It completes one entire 
revolution in its own plane in about 100,000 years (111,149.) 

The mean longitude of the perihelion at the commencement of 
the present century was 99° 30' 5", and of course in the ninth 
degree of Cancer, a little past the winter solstice. In the year 
1248, the perihelion was at the place of this solstice ; and since the 
increase of longitude is 61. "76 a year, hence, 

61."76 : 1 : : 90° : 5246=the time occupied in passing from the 
first of Aries to the solstice. Hence, 5246—1248=3998, which is 
the time before the Christian era, when the perigee was at the 
first of Aries. But this diners only 6 years from the time of the 
creation of the world, which is fixed by chronologists at 4004 
years A. C. At the period of the creation, therefore, the line of 
the apsides of the earth's orbit, coincided with the line of the 
equinoxes. 

197. The angular distance of a body from its aphelion is called 
its Anomaly ; and the interval between the sun's passing the point 
of the ecliptic corresponding to the earth's aphelion, and return- 
ing to the same point again, is called the anomalistic year. This 
period must be a little longer than the sidereal year, since, in order 
to complete the anomalistic revolution, the sun must traverse an 
arc of ll."66 in addition to 360°. 

Now 360° : 365.256 : : ll."66 : 4m. 44s. • 

198. Since the points of the annual orbit, where the sun is at 
the greatest and least distances from the earth, change their posi- 
tion with respect to the solstices, a slow change is occasioned in 
the duration of the respective seasons. For, let the perihelion 
correspond to the place of the winter solstice, as was the case in 
the year 1248 ; then as the sun moves more rapidly in that part 
of his orbit, the winter months will be shorter than the summer. 



MEAN AND TRUE PLACES OF THE SUN. Ill 

But, again, let the perihelion be at the summer solstice, as it will 
be in the year 6485 # ; then the sun will move most rapidly 
through the summer months, and the winters will be longer than 
the summers. At present the perihelion is so near the winter 
solstice, that, the year being divided into summer and winter by 
the equinoxes, the six winter months are passed over between seven 
and eight days sooner than the summer months. 

MEAN AND TRUE PLACES OF THE SUN. 

199. The Mean Motion of any body revolving in an orbit, is 
that which it would have if, in the same time, it involved uniformly 
in a circle. 

In surveying an irregular field, it is common first to strike out 
some regular figure, as a square or a parallelogram, by running 
long lines, and disregarding many small irregularities in the boun- 
daries of the field. By this process, we obtain an approximation 
to the contents of the field, although we have perhaps thrown out 
several small portions which belong to it, and included a number 
of others which do not belong to it. These being separately esti- 
mated and added to or substracted from our first computation, we 
obtain the true area of the field. In a similar manner, we proceed 
in finding the place of a heavenly body, which moves in an orbit 
more or less irregular. Thus we estimate the sun's distance from 
the vernal equinox for every day of the year at noon, on the 
supposition that he moves uniformly in a circular orbit : this is 
the sun's mean longitude. We then apply to this result various 
corrections for the irregularity of the sun's motions, and thus ob- 
tain the true longitude. 

200. The corrections applied to the mean motions of a heav- 
enly body, in order to obtain its true place, are called Equations. 
Thus the elliptical form of the earth's orbit, the precession of the 
equinoxes, and the nutation of the earth's axis, severally affect 
the place of the sun in his apparent orbit, for which equations are 
applied. In a collection of Astronomical Tables, a large part of 



* Biot. 



112 THE SUN. 

the whole are devoted to this object. They give us the amount 
of the corrections to be applied under all the circumstances and 
constantly varying relations in which the sun, moon, and earth 
are situated with respect to each other. The angular distance of 
the earth or any planet from its aphelion, on the supposition that 
it moves uniformly in a circle, is called its Mean Anomaly : its 
actual distance at the same moment in its orbit is called its True 
Anomaly* 

Thus in figure 44, let AEB represent the orbit of the earth 
having the sun in one of the foci at S. Upon AB describe the 
circle AMB. Let E be the place of the earth in its orbit, and M 
the corresponding place in the circle ; then the angle MCA is the 
mean, and ESA the true anomaly. The difference between the 

Fig. 44. 

M 




mean and true anomaly, MCA— ESA, is called the the Equation of 
the Center, being that correction which depends on the elliptical 
form of the orbit, or on the distance of the center of attraction 
from the center of the figure, that is, on the eccentricity of the 
orbit. It is much the greatest of all the corrections used in finding 
the sun's true longitude, amounting, at its maximum, to nearly two 
degrees (1° 55' 26."8.) 

* In some astronomical treatises, the anomaly is reckoned from the perihelion. 



CHAPTER V. 

OF THE MOON LUNAR GEOGRAPHY* PHASES OF THE MOON— HER 

REVOLUTIONS. 

201. Next to the Sun, the Moon naturally claims our attention. 

The Moon is an attendant or satellite to the earth, around which 
she revolves at the distance of nearly 240,000 miles. Her mean 
horizontal parallax being 57' 09",t consequently, sin. 57' 09" : 
semi-diameter of the earth (3956.2) : : rad. : 238,545. (Art. 87.) 

The moon's apparent diameter is 31' 7", and her real diameter 
2083 miles. For, 

Rad. : 238,545 :: sin. 15' 33|" : 1041.6. = moon's semi-diame- 
ter. (See Fig. 26, p. 71.) 

And, since spheres are as the cubes of the diameters, the vol- 
ume of the moon is T V that of the earth. Her density is nearly 
| (.615) the density of the earth, and her mass (=±\x.6l5) is 
about tV . 



202. The moon shines by reflected light borrowed from the 
sun, and when full, exhibits a disk of silvery brightness, diversi- 
fied by extensive portions partially shaded. By the aid of the 
telescope, we see undoubted signs of a varied surface, composed 
of extensive tracts of level country, and numerous mountains and 
valleys. 

203. The line which separates the enlightened from the dark 
portions of the moon's disk, is called the Terminator. (See Fig. 2. 
Frontispiece.) As the terminator traverses the disk from new to 
full moon, it appears through the telescope exceedingly broken in 



* Selenography is a word more appropriate to a description of the moon, but is not 
perhaps sufficiently familiarized by use. 
t Baily's Astronomical Tables. 

15 



114 THE MOON. 

some parts, but smooth in others, indicating that some portions of the 
lunar surface are uneven while others are level. The broken re- 
gions appear brighter than the smooth tracts. The latter have 
been taken for seas, but it is supposed with more probability that 
they are extensive plains, since they are still too uneven for the 
perfect level assumed by bodies of water. That there are moun- 
tains in the moon, is known by several distinct indications. First, 
when the moon is increasing, certain spots are illuminated sooner 
than the neighboring places, appearing like bright points beyond 
the terminator, within the dark part of the disk. (See Fig. 2. 
Frontispiece.) Secondly, after the terminator has passed over 
them, they project shadows upon the illuminated part of the disk, 
always opposite to the sun, corresponding in shape to the form of 
the mountain, and undergoing changes in length from night to 
night, according as the sun shines upon that part of the moon 
more or less obliquely. Many individual mountains rise to a great 
height in the midst of plains, and there are several very remarka- 
ble mountainous groups, extending from a common center in long 
chains. 

204. That there are also valleys in the moon, is equally evident. 
The valleys are known to be truly such, particularly by the man- 
ner in which the light of the sun falls upon them, illuminating the 
part opposite to the sun while the part adjacent is dark, as is the 
case when the light of a lamp shines obliquely into a china cup. 
These valleys are often remarkably regular, and some of them 
almost perfect circles. In several instances, a circular chain of 
mountains surrounds an extensive valley, which appears nearly 
level, except that a sharp mountain sometimes rises from the cen- 
ter. The best time for observing these appearances is near the 
first quarter of the moon, when half the disk is enlightened ;* 
but in studying the lunar geography, it is expedient to observe the 
moon every evening from new to full, or rather through her en- 
tire series of changes. 



* It is earnestly recommended to the student of astronomy, to examine the moon re- 
peatedly with the best telescope he can command, using low powers at first, for the 
sake of a better light. 



LUNAR GEOGRAPHY. 115 

205. The various places on the moon's disk have received ap- 
propriate names. The dusky regions, being formerly supposed to 
be seas, were named accordingly ; and other remarkable places 
have each two names, one derived from some well known spot on 
the earth, and the other from some distinguished personage. Thus 
the same bright spot on the surface of the moon is called Mount 
Sinai or Tycho, and another Mount Etna or Copernicus. The 
names of individuals, however, are more used than the others. 
The frontispiece exhibits the telescopic appearance of the full 
moon. A few of the most remarkable points have the following 
names, corresponding to the numbers and letters on the map. (See 
Frontispiece.) 

1. Tycho, A. Mare Humorum, 

2. Kepler, B. Mare Nubium, 

3. Copernicus, C. Mare Imbrium, 

4. Aristarchus, D. Mare Nectaris, 

5. Helicon, E. Mare Tranquilitatis. 

6. Eratosthenes, F. Mare Serenitatis, 

7. Plato, G. Mare Fecunditatis, 

8. Archimedes, H. Mare Crisium. 

9. Eudoxus, 
10. Aristotle, 

206. The method of estimating the height of lunar mountains is 
as follows. 

Let ABO (Fig. 45,) be the illuminated hemisphere of the moon, 
SO a solar ray touching the moon in O, a point in the circle which 
separates the enlightened from the dark part of the moon. All the 
part ODA will be in darkness ; but if this part contains a moun- 
tain MF, so elevated that its summit M reaches to the solar ray 
SOM, the point M will be enlightened. Let E be the place of the 
observer on the earth, the moon being at any elongation from the 
sun, as measured by the angle EOS. Draw the lines EM, EO, 
and CM, C being the center of the moon ; and let FM be the 
height of the mountain. Draw ON perpendicular to EM. The 
line EO being known, and the angle OEM being measured with a 
micrometer, the value of ON, the projection of the lime OM, be- 



116 



THE MOON. 

Fig. 45. 




comes known. Now OM= 



ON 



, and since OEN is a very 
cos. MON J 

small angle, EON may be considered as a right angle ; conse- 

ON 



■90. Therefore OM=- 

cos. (MOE-90) 

That is, the distance between the summit 



quently, MON=MOE 

ON _ ON 

sin. MOE sin. EOS' 
of the mountain and the illuminated part of the moon's disk, is 
equal to the projected distance as measured by the micrometer, 
divided by the sine of the moon's elongation from the sun. 

Suppose the distance OM=nCO, where n represents the frac- 
tion the part OM is of CO as determined by observation. Then, 
CM 2 =C0 2 +OM 2 =C0 2 +ra 2 C0 2 =C0 2 (l+rc 2 ) .-. CM=CO(l+n 2 H 



1) ?= — CO, neglecting the 



.-.CM-CO or FM=CO {Vl+n* 

higher powers of n, which would be of too little value to be w T orth 
taking into the account. The value of n has been found in one 
case equal to T ^, which gives the height of the mountain equal to 
^ij the semi-diameter of the moon, that is, 3} miles. 

When the moon is exactly at quadrature, then EOM becomes a 
right angle, and the value of OM is obtained directly from actual 
measurement ; and having CO and OM, we easily obtain CM and 
of course FM. 



LUNAR GEOGRAPHY. 117 

207. Schroeter, a German astronomer, estimated the heights of 
the lunar mountains by observations on their shadows. He made 
them in some cases as high as -fa of the semi-diameter of the 
moon, that is, about 5 miles. The same astronomer also estimates 
the depths of some of the lunar valleys at more than four miles. 
Hence it is inferred that the moon's surface is more broken and 
irregular than that of the earth, its mountains being higher and its 
valleys deeper in proportion to the size of the moon than those of 
the earth. 

208. Dr. Herschel is supposed also to have obtained decisive 
evidence of the existence of volcanoes in the moon, not only 
from the light afforded by their fires, but also from the formation 
of new mountains by the accumulation of matter where fires had 
been seen to exist, and which remained after the fires were extinct. 

209. Some indications of an atmosphere about the moon have 
been obtained, the most decisive of which are derived from ap- 
pearances of twilight, a phenomenon that implies the presence 
of an atmosphere. Similar indications have been detected, it is 
supposed, in eclipses of the sun, denoting a transparent refracting 
medium encompassing the moon. The lunar atmosphere, how- 
ever, if any exists, is very inconsiderable in extent and density 
compared with that of the earth.* 

210. The improbability of our ever identifying artificial struc- 
tures in the moon may be inferred from the fact that a line one 
mile in length in the moon subtends an angle at the eye of only 
about one second. If, therefore, works of art were to have a suf- 
ficient horizontal extent to become visible, they can hardly be sup- 
posed to attain the necessary elevation, when we reflect that the 
height of the great pyramid of Egypt is less than the sixth part of 
a mile. 

* See Ed. Encyc. II. 598. 



118 THE MOON. 

PHASES OF THE MOON. 

211. The changes of the moon, commonly called her Phases, 
arise from different portions of her illuminated side being turned 
towards the earth at different times. When the moon is first 
seen after the setting sun, her form is that of a bright crescent, 
on the side of the disk next to the sun, while the other portions 
of the disk shine with a feeble light, reflected to the moon from 
the earth. Every night we observe the moon to be further and 
further eastward of the sun, and at the same time the crescent 
enlarges, until, when the moon has reached an elongation from 
the sun of 90°, half her visible disk is enlightened, and she is 
said to be in her first quarter. The terminator, or line which 
separates the illuminated from the dark part of the moon, is con- 
vex towards the sun from the new moon to the first quarter, and 
the moon is said to be horned. The extremities of the crescent 
are called cusps. At the first quarter, the terminator becomes a 
straight line, coinciding with a diameter of the disk ; but after 
passing this point, the terminator becomes concave towards the 
sun, bounding that side of the moon by an elliptical curve, when 
the moon is said to be gibbous. When the moon arrives at the 
distance of 180° from the sun, the entire circle is illuminated, 
and the moon is full. She is then in opposition to the sun, rising 
about the time the sun sets. For a week after the full, the moon 
appears gibbous again, until, having arrived within 90° of the sun, 
she resumes the same form as at the first quarter, being then at 
her third quarter. From this time until new moon, she exhibits 
again the form of a crescent before the rising sun, until approach- 
ing her conjunction with the sun, her narrow thread of light is lost 
in the solar blaze ; and finally, at the moment of passing the sun, 
the dark side is wholly turned towards us and for some time we 
lose sight of the moon. 

The two points in the orbit corresponding to new and full moon 
respectively, are called by the common name of syzygies ; those 
which are 90° from the sun are called quadratures; and the 
points half way between the syzygies and quadratures are called 
octants. The circle which divides the enlightened from the unen- 
lightened hemisphere of the moon, is called the circle of illumina- 



PHASES. 119 

tion ; that wnich divides the hemisphere that is turned towards 
us from the hemisphere that is turned from us, is called the circle 
of the disk. 

212. As the moon is an opake body of a spherical figure, and 
borrows her light from the sun, it is obvious that that half only 
which is towards the sun can be illuminated. More or less of 
this side is turned towards the earth, according as the moon is at 
a greater or less elongation from the sun. The reason of the dif- 
ferent phases will be best understood from a diagram. Therefore 
let T (Fig. 46,) represent the earth, and S the sun. Let A, B, C, 
&c, be successive positions of the moon. At A the entire dark 

Fiff. 46. 




side of the moon being turned towards the earth, the disk would 
be wholly invisible. At B, the circle of the disk cuts off a small 
part of the enlightened hemisphere, which appears in the heavens 
at b, under the form of a crescent. At C, the first quarter, the 
circle of the disk cuts off half the enlightened hemisphere, and the 
moon appears dichotomized at c. In like manner it will be seen 
that the appearances presented at D, E, F, &c, must be those 
represented at d, e,f. 

REVOLUTIONS OF THE MOON. 

213. The moon revolves around the earth from west to east, 
making the entire circuit of the heavens in about 21\ days. 



120 THE MOON. 

The precise law of the moon's motions in her revolution around 
the earth, is ascertained, as in the case of the sun, (Art. 155,) by 
daily observations on her meridian altitude and right ascension. 
Thence are deduced by calculation her latitude and longitude, 
from which we find, that the moon describes on the celestial 
sphere a great circle of which the earth is the center. 

The period of the moon's revolution from any point in the 
heavens round to the same point again, is called a month. A 
sidereal month is the time of the moon's passing from any star, 
until it returns to the same star again. A synodical month* is 
the time from one conjunction or new moon to another. The 
synodical month is about 29£ days, or more exactly, 29d. 12h. 
44m. 2 S .8=29.53 days. The sidereal month is about two days 
shorter, being 27d. 7h. 43m. ll s .5=27.32 days. As the sun and 
moon are both revolving in the same direction, and the sun is 
moving nearly a degree a day, during the 27 days of the moon's 
revolution, the sun must have moved 27°. Now since the moon 
passes over 360° in 27.32 days, her daily motion must be 13° 17'. 
It must therefore evidently take about two days for the moon to 
overtake the sun. The difference between these two periods 
may, however, be determined with great exactness. The mid- 
dle of an eclipse of the sun marks the exact moment of conjunc- 
tion or new moon ; and by dividing the interval between any 
two solar eclipses by the number of revolutions of the moon, or 
lunations, we obtain the precise period of the synodical month. 
Suppose, for example, two eclipses occur at an interval of 1,000 
lunations ; then the whole number of days and parts of a day 
that compose the interval divided by 1,000 will give the exact 
time of one lunation. f The time of the synodical month being 
ascertained, the exact period of the sidereal month may be derived 
from it. For the arc which the' moon describes in order to come 
into conjunction with the sun, exceeds 360° by the space which 

* aw and o8os, implying that the two bodies come together. 

t It might at first view seem necessary to know the period of one lunation before 
we could know the number of lunations in any given interval. This period is known 
very nearly from the interval between one new moon and another. 



REVOLUTIONS. 121 

the sun has passed over since the preceding conjunction, that is, 
in 29.53 days. Therefore, 

365.24 : 360° :: 29.53 : 29°.l=arc which the moon must de- 
scribe more than 360° in order to overtake the sun. Hence, 

13° 17' : ld.::29°.l : 2.21d.=difference between the sidereal 
and synodical months; and 29.53— 2.21=27.32, the time of the 
sidereal revolution. 

214. The moon's orbit is inclined to the ecliptic in an angle of 
about 5° (5° 8' 48"). It crosses the ecliptic in two opposite points 
called her nodes. The amount of inclination is ascertained by 
observations on the moon's latitude when at a maximum, that 
being of course the greatest distance from the ecliptic, and there- 
fore equal to the inclination of the two circles. 

215. The moon, at the same age, crosses the meridian at differ- 
ent altitudes at different seasons of the year. The full moon, for 
example, will appear much further in the south when on the meri- 
dian at one period of the year than at another. This is owing to 
the fact that the moon's path is differently situated with respect to 
the horizon, at a given time of night at different seasons of the 
year. By taking the ecliptic on an artificial globe to represent 
the moon's path, (which is always near it, Art. 214,) and recollect- 
ing that the new moon is seen in the same part of the heavens 
with the sun, and the full moon in the opposite part of the heavens 
from the sun, we shall readily see that in the winter the new 
moons must run low because the sun does, and for a similar rea- 
son the full moons must run high. It is equally apparent that, in 
summer, when the sun runs high, the new moons must cross the 
meridian at a high, and the full moons at a low altitude. This 
arrangement gives us a great advantage in respect to the amount 
of light received from the moon ; since the full moon is longest 
above the horizon during the long nights of winter, when her pre- 
sence is most needed. This circumstance is especially favorable to 
the inhabitants of the polar regions, the moon, when full, travers- 
ing that part of her orbit which lies north of the equator, and of 
course above the horizon of the north pole, and traversing the por- 
tion that lies south of the equator, and below the polar horizon, 

16 



122 THE MOON. 

when new. During the polar winter, therefore, the moon, from 
the first to the last quarter, is commonly above the horizon, while 
the sun is absent ; whereas, during summer, while the sun is pre- 
sent, the moon is above the horizon while describing her first and 
last quadrants. 

216. About the time of the autumnal equinox, the moon when 
near the full, rises about sunset for a number of nights in succes- 
sion ; and as this is, in England, the period of harvest, the phe- 
nomenon is called the Harvest Moon. To understand the reason 
of this, since the moon is never far from the ecliptic, w T e will 
suppose her progress to be in the ecliptic. If the moon moved 
in the equator, then, since this great circle is at right angles to 
the axis of the earth, all parts of it, as the earth revolves, cut the 
horizon at the same constant angle. But the moon's orbit, or 
the ecliptic, which is here taken to represent it, being oblique 
to the equator, cuts the horizon at different angles in different 
parts, as will easily be seen by reference to an artificial globe. 
When the first of Aries, or vernal equinox, is in the eastern hori- 
zon, it will be seen that the ecliptic, (and consequently the moon's 
orbit.) makes its least angle with the horizon. Now at the au- 
tumnal equinox, the sun being in Libra, the moon at the full is in 
Aries, and rises when the sun sets. On the following evening, 
although she has advanced in her orbit about 13°, (Art. 213,) yet 
her progress being oblique to the horizon, and at a small angle 
with it, she will be found at this time but a little way below the 
horizon, compared with the point where she was at sunset the 
preceding evening. She therefore rises but little later, and so 
for a week only a little later each evening than she did the pre- 
ceding night. 

217. The moon is about ^\ nearer to us when near the zenith 
than when in the horizon. 

The horizontal distance CD (Fig. 47,) is nearly equal to AD= 
AD', which is greater than CD' by AC, the semi-diameter of the 
earth =eV the distance of the moon. Accordingly, the apparent 
diameter of the moon, when actually measured, is about 30" 
(which equals about ? V of the whole) greater when in the zenith 



REVOLUTIONS. 



123 



than in the horizon. The apparent enlargement of the full moon 
when rising, is owing to the same causes as that of the sun, as ex- 
plained in article 96. 

Fig. 47. 




218. The moon turns on its axis in the same time in which it 
revolves around the earth. 

This is known by the moon's always keeping nearly the same 
face towards us, as is indicated by the telescope, which could not 
happen unless her revolution on her axis kept pace with her mo- 
tion in her orbit. Thus, it will be seen by inspecting figure 31, 
that the earth turns different faces towards the sun at different 
times ; and if a ball having one hemisphere white and the other 
black be carried around a lamp, it will easily be seen that it can- 
not present the same face constantly towards the lamp unless it 
turns once on its axis while performing its revolution. The same 
thing will be observed when a man walks around a tree, keeping 
his face constantly towards it. Since however the motion of the 
moon on its axis is uniform, while the motion in its orbit is une- 
qual, the moon does in fact reveal to us a little sometimes of one 
side and sometimes of the other. Thus when the ball above 
mentioned is placed before the eye with its light side towards us, 
or carrying it round, if it is moved faster than it is turned on its 
axis, a portion of the dark hemisphere is brought into view on 
one side ; or if it is moved forward slower than it is turned on 
its axis, a portion of the dark hemisphere comes into view on the 
other side. 

219. These appearances are called the moon's librations in lon- 
gitude. The moon has also a libration in latitude, so called, be- 
cause in one part of her revolution, more of the region around one 



124 THE MOON. 

of the poles comes into view, and in another part of the revolu- 
tion, more of the region around the other pole ; which gives the ap- 
pearance of a tilting motion to the moon's axis. This has nearly the 
same cause with that which occasions our change of seasons. The 
moon's axis being inclined to that of the ecliptic, about 1^ degrees, 
(1° 30' 10".8,) and always remaining parallel to itself, the circle 
which divides the visible from the invisible part of the moon, will 
pass in such a way as to throw sometimes more of one pole into 
view and sometimes more of the other, as would be the case with 
the earth if seen from the sun. (See Fig. 31.) 

The moon exhibits another phenomenon of this kind called 
her diurnal librarian, depending on the daily rotation of the 
spectator. She turns the same face towards the center of the 
earth only, whereas we view her from the surface. When she is 
on the meridian, we see her disk nearly as though we viewed it 
from the center of the earth, and hence in this situation it is sub- 
ject to little change ; but when near the horizon, our circle of 
vision takes in more of the upper limb than would be presented 
to a spectator at the center of the earth. Hence, from this cause, 
we see a portion of one limb while the moon is rising, which is 
gradually lost sight of, and we see a portion of the opposite limb 
as the moon declines towards the west. It will be remarked that 
neither of the foregoing changes implies any actual motion in the 
moon, but that each arises from a change of position in the spec- 
tator relative to the moon. 

220. An inhabitant of the moon would have but one day and 
one night during the whole lunar month of 29| days. One of 
its days, therefore, is equal to nearly 15 of ours. So protracted 
an exposure to the sun's rays, especially in the equatorial regions 
of the moon, must occasion an excessive accumulation of heat ; 
and so long an absence of the sun must occasion a corresponding 
degree of cold. Each day would be a wearisome summer ; each 
night a severe winter.* A spectator on the side of the moon 
which is opposite to us would never see the earth ; but one on the 
side next to us would see the earth presenting a gradual succession 

* Francoeur, Uranog. p. 91. 



REVOLUTIONS. 125 

of changes during his long night of 360 hours. Soon after the 
earth's conjunction with the sun, he would have the light of the 
earth reflected to him, presenting at first a crescent, but enlarging, 
as the earth approaches its opposition, to a great orb, 13 times as 
large as the full moon appears to us, and affording nearly 13 times 
as much light. Our seas, our plains, our mountains, our volcanoes, 
and our clouds, would produce very diversified appearances, as 
would the various parts of the earth brought successively into 
view by its diurnal rotation. The earth while in view to an in- 
habitant of the moon, would remain immovably fixed in the same 
part of the heavens. For being unconscious of his own motion 
around the earth, as we are of our motion around the sun, the 
earth would seem to revolve around his own planet from west to 
east ; but, meanwhile, his rotation along with the moon on her 
axis, would cause the earth to have an apparent motion westward 
at the same rate. The two motions, therefore, would exactly 
balance each other, and the earth would appear all the while at 
rest. « The earth- is full to the moon when the latter is new to us ; 
and universally the two phases are complementary to each other.* 

221. It has been observed already, (Art. 214,) that the moon's 
orbit crosses the ecliptic in two opposite points called the nodes. 
That which the moon crosses from south to north, is called the 
ascending node ; that which the moon crosses from north to south, 
the descending node. 

From the manner in which the figure representing the earth's 
orbit and that of the moon, is commonly drawn, the learner is 
sometimes puzzled to see how the orbit of the moon can cut the 
ecliptic in two points directly opposite to each other. But he must 
reflect that the lunar orbit cuts the plane of the ecliptic and not 
the earth's path in that plane, although these respective points are 
projected upon that path in the heavens. 

222. We have thus far contemplated the revolution of the moon 
around the earth as though the earth were at rest. But, in order 
to have just ideas respecting the moon's motions, we must recol- 
lect that the moon likewise revolves along with the earth around 

* Francoeur, p. 92. 



126 THE MOON. 

the sun. It is sometimes said that the earth carries the moon 
along with her in her annual revolution. This language may- 
convey an erroneous idea ; for the moon, as well as the earth, 
revolves around the sun under the influence of two forces, and 
would continue her motion around the sun, were the earth re- 
moved out of the way. Indeed, the moon is attracted towards 
the sun 2} times more than towards the earth,* and would aban- 
don the earth were not the latter also carried along with her by 
the same forces. So far as the sun acts equally on both bodies, 
their motion with respect to each other would not be disturbed. 
Because the gravity of the moon towards the sun is found to be 
greater, at the conjunction, than her gravity towards the earth, 
some have apprehended that, if the doctrine of universal gravi- 
tation is true, the moon ought necessarily to abandon the earth. 
In order to understand the reason why it does not do thus we 
must reflect, that when a body is revolving in its orbit under the 
action of the projectile force and gravity, whatever diminishes 
the force of gravity while that of projection remains the same, 
causes the body to recede from the center; and whatever in- 
creases the amount of gravity carries the body towards the center. 
Now, when the moon is in conjunction, her gravity towards the 
earth acts in opposition to that towards the sun, while her velocity 
remains too great to carry her, with what force remains, in a 
circle about the sun, and she therefore recedes from the sun, and 
commences her revolution around the earth. On arriving at the 
opposition, the gravity of the earth conspires with that of the sun, 
and the moon's projectile force being less than that required to 
make her revolve in a circular orbit, when attracted towards the 
sun by the sum of these forces, she accordingly begins to approach 
the sun and descends again to the conjunction.! 



* It is shown by writers on Mechanics, that the forces with which bodies revolving 
in circular orbits tend towards their centers, are as the radii of their orbits divided 
by the squares of their periodical times. Hence, supposing the orbits of the earth and 
the moon to be circular, (and their slight eccentricity will not much affect the re- 
suit,) we have 

400 1 

G : G ' : : (365.25)2 : (27.32) 2: : 2 " 2 : 1 ". 
t M'Laurin's Discoveries of Newton, B. iv, ch. 5. 



LUNAR IRREGULARITIES. 127 

223. The attraction of the sun, however, being every where 
greater than that of the earth, the actual path of the moon around 
the sun is every where concave towards the latter. Still the el- 
liptical path of the moon around the earth, is to be conceived of 
in the same way as though both bodies were at rest with respect 
to the sun. Thus, while a steamboat is passing swiftly around an 
island, and a man is walking slowly around a post in the cabin, 
the line which he describes in space between the forward motion 
of the boat and his circular motion around the post, may be every 
where concave towards the island, while his path around the post 
will still be the same as though both were at rest. A nail in the 
rim of a coach wheel, will turn around the axis of the wheel, when 
the coach has a forward motion in the same manner as when the 
coach is at rest, although the line actually described by the nail 
will be the resultant of both motions, and very different from 
either. 



CHAPTER VI 



LUNAR IRREGULARITIES. 



224. We have hitherto regarded the moon as describing a great 
circle on the face of the sky, such being the visible orbit as seen 
by projection. But, on more exact investigation, it is found that 
her orbit is not a circle, and that her motions are subject to very- 
numerous irregularities. These will be best understood in con- 
nection with the causes on which they depend. The law of uni- 
versal gravitation has been applied with wonderful success to their 
investigation, and its results have conspired with those of long 
continued observation, to furnish the means of ascertaining with 
great exactness the place of the moon in the heavens at any given 
instant of time, past or future, and thus to enable astronomers to 
determine longitudes, to calculate eclipses, and to solve various 
other problems of the highest interest. A complete understand- 
ing of all the irregularities of the moon's motions, must be sought 



128 THE MOON. 

for in more extensive treatises of astronomy than the present ; but 
some general acquaintance with the subject, clear and intelligible 
as far as it goes, may be acquired by first gaining a distinct idea 
of the mutual actions of the sun, the moon, and the earth. 

225. The irregularities of the moon's motions, are due chiefly to 
the disturbing influence of the sun, which operates in two ways ; first, 
by acting unequally on the earth and moon, and, secondly, by acting 
obliquely on the moon, on account of the inclination of her orbit to 
the ecliptic* 

If the sun acted equally on the earth and moon, and always in 
parallel lines, this action would serve only to restrain them in their 
annual motions round the sun, and would not affect their actions 
on each other, or their motions about their common center of 
gravity. In that case, if they were allowed to fall directly to- 
wards the sun, they would fall equally, and their respective situa- 
tions would not be affected by their descending equally towards 
it. We might then conceive them as in a plane, every part of 
which being equally acted on by the sun, the whole plane would 
descend towards the sun, but the respective motions of the earth 
and the moon in this plane, would be the same as if it were qui- 
escent. Supposing then this plane and all in it, to have an annual 
motion imprinted on it, it would move regularly round the sun, 
while the earth and moon would move in it with respect to each 
other, as if the plane were at rest, without any irregularities. 
But because the moon is nearer the sun in one half of her orbit 
than the earth is, and in the other half of her orbit is at a greater 
distance than the earth from the sun, while the power of gravity 
is always greater at a less distance ; it follows, that in one half of 
her orbit the moon is more attracted than the earth towards the 
sun, and in the other half less attracted than the earth. The ex- 
cess of the attraction, in the first case, and the defect in the second, 
constitutes a disturbing force, to which we may add another, 
namely, that arising from the oblique action of the solar force, 
since this action is not directed in parallel lines, but in lines that 
meet in the center of the sun. 

* M'Laurin's Discoveries of Newton, B. iv, ch. 4. La Place's Syst. du Monde, 
B. iv, ch. 5. 



LUNAR IRREGULARITIES. 129 

226. To see the effects of this process, let us suppose that the 
projectile motions of the earth and moon were destroyed, and 
that they were allowed to fall freely towards the sun. If the 
moon was in conjunction with the sun, or in that part of her orbit 
which is nearest to him, the moon would be more attracted than 
the earth, and fall with greater velocity towards the sun ; so that 
the distance of the moon from the earth would be increased in the 
fall. If the moon was in opposition, or in the part of her orbit 
which is furthest from the sun, she would be less attracted than 
the earth by the sun, and would fall with a less velocity towards 
the sun, and would be left behind ; so that the distance of the 
moon from the earth would be increased in this case also. If the 
moon was in one of the quarters, then the earth and moon being 
both attracted towards the center of the sun, they would both de- 
scend directly towards that center, and by approaching it, they 
would necessarily at the same time approach each other, and in 
this case their distance from each other would be diminished. 
Now whenever the action of the sun would increase their distance, 
if they were allowed to fall towards the sun, then the sun's action, 
by endeavoring to separate them, diminishes their gravity to each 
other ; whenever the sun's action would diminish the distance, then 
it increases their mutual gravitation. Hence, in the conjunction 
and opposition, that is, in the syzygies, their gravity towards each 
other is diminished by the action of the sun, while in the quadra- 
tures it is increased. But it must be remembered that it is not 
the total action of the sun on them that disturbs their motions, 
but only that part of it which tends at one time to separate them, 
and at another time to bring them nearer together. The other 
and far greater part, has no other effect than to retain them in 
their annual course around the sun. 

227. Suppose the moon setting out from the quarter that pre- 
cedes the conjunction with a velocity that would make her de- 
scribe an exact circle round the earth, if the sun's action had no 
effect on her : since her gravity is increased by that action, she mus* 
descend towards the earth and move within that circle. Her 01 
bit then would be more curved than it otherwise would have been ; 
because the addition to her gravity will make her fall further at 

17 



130 



THE MOON. 



the end of an arc below the tangent drawn at the other end of it. 
Her motion will be thus accelerated, and it will continue to be 
accelerated until she arrives at the ensuing conjunction, because 
the direction of the sun's action upon her, during that time, makes 
an acute angle with the direction of her motion. (See Fig. 41.) 
At the conjunction, her gravity towards the earth being diminished 
by the action of the sun, her orbit will then be less curved, and 
she will be carried further from the earth as she moves to the next 
quarter ; and because the action of the sun makes there an obtuse 
angle with the direction of her motion, she will be retarded in the 
same degree as she was accelerated before. 



228. After this general explanation of the mode in which the 
sun acts as a disturbing force on the motions of the moon, the 
learner will be prepared to understand the mathematical develop- 
ment of the same doctrine. 

Therefore, let ADBC (Fig. 48,) be the orbit, nearly circular, in 
which the moon M revolves in the direction CADB, round the 



earth E. Let S be the sun, and let 
SE the radius of the earth's orbit, 
be taken to represent the force with 
which the earth gravitates to the sun. 
1 1 



Fig. 48. 



Then (Art. 180,) 



SE 2 *SM 2 



SE : 



SE* 



SM : 



^ = the force by which the sun 



draws the moon in the direction 

SE 3 



MS. Take MG 



SM 2 ' 



and let the 



parallelogram KF be described, 
having MG for its diagonal, and 
having its sides parallel to EM and 
ES. The force MG may be re- 
solved into two, MF and MK, of 
which MF, directed towards E, the 
center of the earth, increases the 

gravity of the moon to the earth, and does not hinder the areas 
described by the radius vector from being proportional to the 




LUNAR IRREGULARITIES. ]31 

times. The other force MK draws the moon in the direction of 
the line joining the centers of the sun and earth. It is, however, 
only the excess of this force, above the force represented by SE, 
or that which draws the earth to the sun, which disturbs the rela- 
tive position of the moon and earth. This is evident, for if KM 
were just equal to ES, no disturbance of the moon relative to the 
earth could arise from it. If then ES be taken from MK, the dif- 
ference HK is the whole force in the direction parallel to SE, by 
which the sun disturbs the relative position of the moon and earth. 
Now, if in MK, MN be taken equal to HK, and if NO be drawn 
perpendicular to the radius vector EM produced, the force MN 
may be resolved into two, MO and ON, the first lessening the 
gravity of the moon to the earth ; and the second, being parallel 
to the tangent of the moon's orbit in M, accelerates the moon's 
motion from C to A, and retards it from A to D, and so alternately 
in the other two quadrants. Thus the whole solar force directed 
to the center of the earth, is composed of the two parts MF and 
MO, which are sometimes opposed to one another, but which 
never affect the uniform description of the areas about E. Near 
the quadratures the force MO vanishes, and the force MF, which 
increases the gravity of the moon to the earth, coincides with CE 
or DE. As the moon approaches the conjunction at A, the force 
MO prevails over MF, and lessens the gravity of the moon to the 
earth. In the opposite point of the orbit, when the moon is in op- 
position at B, the force with which the sun draws the moon is less 
than that with which the sun draws the earth, so that the effect of 
the solar force is to separate the moon and earth, or to increase 
their distance ; that is, it is the same as if, conceiving the earth 
not to be acted on, the sun's force drew the moon in the direction 
from E to B. This force is negative, therefore, in respect to the 
force at A, and the effect in both cases is to draw the moon from 
the earth in a direction perpendicular to the line of the quadra- 
tures. Hence, the general result is, that by the disturbing force 
of the sun, the gravity to the earth is increased at the quadratures, 
and diminished at the syzygies. It is found by calculation that the 
average amount of this disturbing force is jj ¥ of the moon's 
gravity to the earth.* 

* Playfair. 



132 THE MOON. 

229. With these general principles in view, we may now pro- 
ceed to investigate the figure of the moon's orbit, and the irregu- 
larities to which the motions of this body are subject. 

230. The figure of the moon's orbit is an ellipse, having the earth 
in one of the foci. 

The elliptical figure of the moon's orbit, is revealed to us by ob- 
servations on her changes in apparent diameter, and in her hori- 
zontal parallax. First, we may measure from day to day the ap- 
parent diameter of the moon. Its variations being inversely as 
the distances, (Art. 163,) they give us at once the relative distance 
of each point of observation from the focus. Secondly, the va- 
riations on the moon's horizontal parallax, which also are inversely 
as the distances, (Art. 82,) lead to the same results. Observations 
on the angular velocities, combined with the changes in the lengths 
of the radius vector, afford the means of laying down a plot of the 
lunar orbit, as in the case of the sun, represented in figure 32. 
The orbit is shown to be nearly an ellipse, because it is found to 
have the properties of an ellipse. 

The moon's greatest and least apparent diameters are respectively 
33'.518 and 29'. 365, while her corresponding changes of parallax 
are 61'.4 and 53'. 8. The two ratios ought to be equal, and we 
shall find such to be the fact very nearly, as expressed by the fore- 
going numbers ; for, 

61.4 : 53.8 : : 33.518 : 29.369. 

The greatest and least distances of the moon from the earth, 
derived from the parallaxes, are 63.8419 and 55.9164, or nearly 
64 and 56. the radius of the earth being taken for unity. Hence, 
taking the arithmetical mean, which is 59.879, we find that the 
mean distance of the moon from the earth is very nearly 60 times 
the radius of the earth. The point in the moon's orbit nearest 
the earth, is called her perigee ; the point furthest from the earth, 
her apogee. 

The greatest and least apparent diameters of the sun are re- 
spectively 32.583, and 31.517, which numbers express also the ratio 
of the greatest and least distances of the earth from the sun. By 
comparing this ratio with that of the distances of the moon, it will 
be seen that the latter vary much more than the former, and con- 



LUNAR IRREGULARITIES. 133 

sequently that the lunar orbit is much more eccentric than the so- 
lar. The eccentricity of the moon's orbit is in fact 0.0548, (the 
semi-major axis being as usual taken for unity) = T \ of its mean 
distance from the earth, while that of the earth is only .01685—V 
of its mean distance from the sun. 



231. The mooris nodes constantly shift their positions in the eclip- 
tic from east to west, at the rate of 19° 35' per annum, returning to 
the same points in 18.6 years. 

Suppose the great circle of the ecliptic marked out on the face 
of the sky in a distinct line, and let us observe, at any given time, 
the exact point where the moon crosses this line, which we will 
suppose to be close to a certain star ; then, on its next return to 
that part of the heavens, we shall find that it crosses the ecliptic 
sensibly to the westward of that star, and so on, further and fur- 
ther to the westward every time it crosses the ecliptic at either 
node. This fact is expressed by saying that the nodes retrograde 
on the ecliptic, and that the line which joins them, or the line of 
the nodes, revolves from east to west. 

232. This shifting of the moon's nodes implies that the lunar 
orbit is not a curve returning into itself, but that it more resem- 
bles a spiral like the curve represented in figure 49, where ahc 
represents the ecliptic, and ABC the 
lunar orbit, having its nodes at C and 
E, instead of A and a ; consequently, 
the nodes shift backwards through 
the arcs aC and AE. The manner 
in which this effect is produced may 
be thus explained. That part of the 

solar force which is parallel to the line joining the centers of the 
sun and earth, (See Fig. 48,) is not in the plane of the moon's 
orbit, (since this is inclined to the ecliptic about 5°,) except when 
the sun itself is in that plane, or when the line of the nodes being 
produced, passes through the sun. In all other cases it is oblique 
to the plane of the orbit, and may be resolved into two forces, 
one of which is at right angles to that plane, and is directed to- 
wards the ecliptic. This force of course draws the moon continu- 




134 



THE MOON. 



ally towards the ecliptic, or produces a continual deflection of the 
moon from the plane of her own orbit towards that of the earth. 
Hence the moon meets the plane of the ecliptic sooner than it 
would have done if that force had not acted. At every half revo- 
lution, therefore, the point in which the moon meets the ecliptic, 
shifts in a direction contrary to that of the moon's motion, or con- 
trary to the order of the signs. If the earth and sun were at rest, 
the effect of the deflecting force just described, would be to pro- 
duce a retrograde motion of the line of the nodes till that line was 
brought to pass through the sun, and of consequence, the plane of 
the moon's orbit to do the same, after which they would both re- 
main in their position, there being no longer any force tending to 
produce change in either. But the motion of the earth carries the 
line of the nodes out of this position, and produces, by that means, 
its continual retrogradation. The same force produces a small 
variation in the inclination of the moon's orbit, giving it an alter- 
nate increase and decrease within very narrow limits.* These 
points will be easily understood by the aid of a diagram. There- 
fore, let MN (Fig. 50,) be the ecliptic, ANB the orbit of the moon, 
the moon being in L, and N its descending node. Let the disturb- 
ing force of the sun which tends to bring it down to the ecliptic 

Fig. 50. 




be represented by U, and its velocity in its orbit by ha. Under 
the action of these two forces, the moon will describe the diago- 
nal Lc of the parallelogram ba, and its orbit will be changed from 
AN to LN' ; the node N passes to N' ; and the exterior angle at N' 
of the triangle LNN' being greater than the interior and opposite 



Playfair. 



LUNAR IRREGULARITIES. 135 

angle at N, the inclination of the orbit is increased at the node. 
After the moon has passed the ecliptic to the south side to Z, the 
disturbing force Id produces a new change of the orbit N'Ze to 
W'lf, and the inclination is diminished as at N". In general, 
while the moon is receding from one of the nodes, its inclination is 
diminishing : while it is approaching a node, the inclination is in- 
creasing.* 

233. The period occupied by the sun in passing from one of 
the moon's nodes until it comes round to the same node again, is 
called the synodical revolution of the node. This period is shorter 
than the sidereal year, being only about 346i days. For since 
the node shifts its place to the westward 19° 35' per annum, the 
sun, in his annual revolution, comes to it so much before he com- 
pletes his entire circuit ; and since the sun moves about a degree 
a day, the synodical revolution of the node is 365— 19 ==346, or 
more exactly, 346.619851. The time from one new moon, or 
from one full moon, to another, is 29.5305887 days. Now 19 
synodical revolutions of the nodes contain very nearly 223 of 
these periods. 

For 346.619851 x 19=6585.78, 

And 29.5305887x223=6585.32. 
Hence, if the sun and moon were to leave the moon's node toge- 
ther, after the sun had been round to the same node 19 times, the 
moon would have performed very nearly 223 synodical revolu- 
tions, and would, therefore, at the end of this period meet at the 
same node, to repeat the same circuit. And since eclipses of the 
sun and moon depend upon the relative position of the sun, the 
moon, and node, these phenomena are repeated in nearly the same 
order, in each of those periods. Hence, this period, consisting of 
about 18 years and 10 days, under the name of the Saros, was 
used by the Chaldeans and other ancient nations in predicting 
eclipses. 

234. The Metonic Cycle is not the same with the Saros, but 
consists of 19 tropical years. During this period the moon makes 



* Francoeur, Uranog. p. 158.— Robison's Phys. Astronomy, Art. 264. 



136 THE MOON. 

very nearly 235 synodical revolutions, and hence the new and full 
moons, if reckoned by periods of 19 years, recur at the same 
dates. If, for example, a new moon fell on the fiftieth day of one 
cycle, it would also fall on the fiftieth day of each succeeding cycle ; 
and, since the regulation of games, feasts, and fasts, has been 
made very extensively according to new or full moons, hence this 
lunar cycle has been much used both in ancient and modern 
times. The Athenians adopted it 433 years before the Christian 
era, for the regulation of their calendar, and had it inscribed in 
letters of gold on the walls of the temple of Minerva. Hence the 
term Golden Number, which denotes the year of the lunar cycle. 

235. The line of the apsides of the mooris orbit revolves from 
west to east through her whole orbit in about nine years. 

If, in any revolution of the moon, we should accurately mark 
the place in the heavens where the moon comes to its perigee, 
(Art. 230,) we should find, that at the next revolution, it would 
come to its perigee at a point a little further eastward than before, 
and so on at every revolution, until, after 9 years, it would come 
to its perigee at nearly the same point as at first. This fact is 
expressed by saying that the perigee, and of course the apogee, 
revolves, and that the line which joins these two points, or the line 
of the apsides, also revolves. 

The place of the perigee may be found by observing when the 
moon has the greatest apparent diameter. But as the magnitude 
of the moon varies slowly at this point, a better method of ascer- 
taining the position of the apsides, is to take two points in the or- 
bit where the variations in apparent diameter are most rapid, and 
to find where they are equal on opposite sides of the orbit. The 
middle point between the two will give the place of the perigee. 

The angular distance of the moon from her perigee in any part 
of her revolution, is called the Moon's Anomaly. 

* 

236. The change of place in the apsides of the moon's orbit, 
like the shifting of the nodes, is caused by the disturbing influence 
of the sun. If when the moon sets out from its perigee, it were 
urged by no other force than that of projection, combined with its 
gravitation towards the earth, it would describe a symmetrical 



LUNAR IRREGULARITIES. 137 

curve (Art. 186,) coming to its apogee at the distance of 180°. 
But as the mean disturbing force in the direction of the radius 
vector tends, on the whole, to diminish the gravitation of the 
moon to the earth, the portion of the path described in an instant 
will be less deflected from her tangent, or less curved, than if this 
force did not exist. Hence the path of the moon will not inter- 
sect the radius vector at right angles, that is, she will not arrive at 
her apogee until after passing more than 180° from her perigee, 
by which means these points will constantly shift their positions 
from west to east.* The motion of the apsides is found to be 3° 
1' 20" for every sidereal revolution of the moon. 

237. On account of the greater eccentricity of the moon's orbit 
above that of the sun, the Equation of the Center, or that correc- 
tion which is applied to the moon's mean anomaly to find her true 
anomaly (Art. 200,) is much greater than that of the sun, being 
when greatest more than six degrees, (6° 17' 12". 7,) while that of 
the sun is less than two degrees, (1° 55' 26".8.) 

The irregularities in the motions of the moon may be compared 
to those of the magnetic needle. As a first approximation, we say 
that the needle places itself in a north and south line. On closer 
examination, however, we find that, at different places, it deviates 
more or less from this line, and we introduce the first great cor- 
rection under the name of the declination of the needle. But ob- 
servation shows us that the declination alternately increases and 
diminishes every day, and therefore we apply to the declination 
itself a second correction for the diurnal variation. Finally, we 
ascertain, from long continued observations, that the diurnal va- 
riation is affected by the change of seasons, being greater in sum- 
mer than in winter, and hence we apply to the diurnal variation a 
third correction for the annual variation. 

In like manner, we shall find the greater inequalities of the 
moon's motions are themselves subject to subordinate inequalities, 
which give rise to smaller equations, and these to smaller still, to 
the last degree of refinement. 

238. Next to the equation of the center, the greatest correction 

1 o * Playfair. 



138 THE MOON. 

to be applied to the moon's longitude, is that which belongs to the 
Evection. The evection is a change of form in the lunar orbit, by 
which its eccentricity is sometimes increased, and sometimes 
diminished. It depends on the position of the line of the apsides 
with respect to the line of the syzygies. 

This irregularity, and its connexion with the place of the peri- 
gee with respect to the place of conjunction or opposition, was 
known as a fact to the ancient astronomers, Hipparchus and 
Ptolemy ; but its cause was first explained by Newton in con- 
formity with the law of universal gravitation. It was found, by 
observation, that the equation of the center itself was subject to a 
periodical variation, being greater than its mean, and greatest of 
all when the conjunction or opposition takes place at the perigee 
or apogee, and ieast of all when the conjunction or opposition 
takes place at a point half way between the perigee and apogee ; 
or, in the more common language of astronomers, the equation of 
the center is increased when the line of the apsides is in syzygy, 
and diminished when that line is in quadrature. If, for example, 
when the line of the. apsides is in syzygy, we compute the moon's 
place by deducting the equation of the center from the mean 
anomaly (see Art. 200,) seven days after conjunction, the compu- 
ted longitude will be greater than that determined by actual obser- 
vation, by about 80 minutes ; but if the same estimate is made 
when the line of the apsides is in quadrature, the computed longi- 
tude will be less than the observed, by the same quantity. These 
results plainly show a connexion between the velocity of the 
moon's motions and the position of the line of the apsides with 
respect to the line of the syzygies. 

239. Now any cause which, at the , perigee, should have the 
effect to increase the moon's gravitation towards the earth beyond 
its mean, and, at the apogee, to diminish the moon's gravitation 
towards the earth, would augment the difference between the 
gravitation at the perigee and at the apogee, and consequently in- 
crease the eccentricity of the orbit. Again, any cause which at 
the perigee should have the effect to lessen the moon's gravitation 
towards the earth, and, at the apogee, to increase it, would lessen 
the difference between the two, and consequently diminish the 



LUNAR IRREGULARITIES. 139 

eccentricity of the orbit, or bring it nearer to a circle. Let us 
see if the disturbing force of the sun produces these effects. The 
sun's disturbing force, as we have seen in article 228, admits of 
two resolutions, one in the direction of the radius vector, (OM, 
Fig. 48,) the othej; (ON) in the direction of a tangent to the orbit. 
First, let AB be the line of the apsides in syzygy, A being the place 
of the perigee. The sun's disturbing force OM is greatest in the 
direction of the line of the syzygies ; yet depending as it does on the 
unequal action of the sun upon the earth and the moon, and being 
greater as their distance from each other is greater, it is at a mini- 
mum when acting at the perigee, and at a maximum when acting at 
the apogee. Hence its effect is to draw away the moon from the 
earth less than usual at the perigee, and of course to make her 
gravitation towards the earth greater than usual, while at the 
apogee its effect is to diminish the tendency of the moon to the 
earth more than usual, and thus to increase the disproportion be- 
tween the two distances of the moon from the focus at these two 
points, and of course to increase the eccentricity of the orbit. 
The moon, therefore, if moving towards the perigee, is brought 
to the line of the apsides in a point between its mean place and 
the earth ; or if moving towards the apogee, she reaches the line 
of the apsides in a point more remote from the earth than its mean 
place. 

Secondly, let CD be the line of the apsides, in quadrature, C 
being the place of the perigee. The effect of the sun's disturb- 
ing force is to increase the tendency of the moon towards the 
earth when in quadrature. If, however, the moon is then at her 
perigee, such increase will be less than usual, and if at her apogee, 
it will be more than usual ; hence its effect will be to lessen the 
disproportion between the two distances of the moon from the 
focus at these two points ; and of course to diminish the eccen- 
tricity of the orbit. The moon, therefore, if moving towards 
the perigee, meets the line of the apsides in a point more remote 
from the earth than the mean place of the perigee ; and if moving 
towards the apogee, in a point between the earth and the mean place 
of the apogee.* 

* Woodhousc's Ast. p. 680. 



140 THE MOON. 

240. A third inequality in the lunar motions, is the Variation. 
By comparing the moon's place as computed from her mean mo- 
tion corrected for the equation of the center and for evection, 
with her place as determined by observation, Tycho Brahe dis- 
covered that the computed and observed places did not always 
agree. They agreed only in the syzygies and quadratures, and 
disagreed most at a point half way between these, that is, at the 
octants. Here, at the maximum, it amounted to more than half 
a degree (35' 41. "6.) It appeared evident from examining the 
daily observations while the moon is performing her revolution 
around the earth, that this inequality is connected with the angular 
distance of the moon from the sun, and in every part of the orbit 
could be correctly expressed by multiplying the maximum value 
as given above, into the sine of twice the angular distance between 
the sun and the moon. It is, therefore, at the conjunctions and 
quadratures, and greatest at the octants. Tycho Brahe knew the 
fact : Newton investigated the cause. 

It appears by article 228, that the sun's disturbing force can be 
resolved into two parts, — one in the direction of the radius vector, 
the other at right angles to it in the direction of a tangent to the 
moon's orbit. The former, as already explained, produces the 
Evection: the latter produces the Variation. This latter force 
will accelerate the moon's velocity, in every point of the quadrant 
w T hich the moon describes in moving from quadrature to conjunc- 
tion, or from C to A, (Fig. 48,) but at an unequal rate, the 
acceleration being greatest at the octant, and nothing at the quad- 
rature and the conjunction ; and when the moon is past conjunction, 
the tangential force will change its direction and retard the moon's 
motion. All these points will be understood by inspection of 
figure 48. 

241. A fourth lunar inequality is the Annual Equation. This 
depends on the distance of the earth (and of course the moon) 
from the sun. Since the disturbing influence of the sun has a 
greater effect in proportion as the sun is nearer,* consequently all 
the inequalities depending on this influence must vary at different 

* Varying reciprocally as the cube of the sun's distance from the earth. 



LTJNAR IRREGULARITIES. 141 

seasons of the year. Hence, the amount of this effect due to any- 
particular time of the year is called the Annual Equation. 

242. The foregoing are the largest of the inequalities of the 
moon's motions, and may serve as specimens of the corrections that 
are to be applied to the mean place of the moon in order to find 
her true place. These were first discovered by actual observa- 
tion ; but a far greater number, though most of them are exceed- 
ingly minute, have been made known by the investigations of Phys- 
ical Astronomy, in following out all the consequences of universal 
gravitation. In the best tables, about 30 equations are applied to 
the mean motions of the moon. That is, we first compute the 
place of the moon on the supposition that she moves uniformly 
in a circle. This gives us her mean place. We then proceed, 
by the aid of the Lunar Tables, to apply the different corrections, 
such as the equation of the center, evection, variation, the annual 
equation, and so on, to the number of 28. Numerous as these 
corrections appear, yet La Place informs us, that the whole num- 
ber belonging to the moon's longitude is no less than 60 ; and 
that to give the tables all the requisite degree of precision, addi- 
tional investigations will be necessary, as extensive at least as 
those already made.* The best tables in use in the time of Tycho 
Brahe, gave the moon's place only by a distant approximation. 
The tables in use in the time of Newton, (Halley's tables,) approxi- 
mated within 7 minutes. Tables at present in use give the moon's 
place to 5 seconds. These additional degrees of accuracy have 
been attained only by immense labor, and by the united efforts of 
Physical Astronomy and the most refined observations. 

243. The inequalities of the moon's motions are divided into 
periodical and secular. Periodical inequalities are those which 
are completed in comparatively short periods, like evection and 
variation: Secular inequalities are those which are completed 
only in very long periods, such as centuries or ages. Hence the 
corresponding terms periodical equations, and secular equations. 
As an example of a secular inequality, we may mention the ac- 

* Syst. du Monde, I. iv, c. 5. 



142 THE MOON. 

celeration of the moon's mean motion. It is discovered, that the 
moon actually revolves around the earth in less time now than 
she did in ancient times. The difference however is exceedingly 
small, being only about 10" in a century, but increases from century 
to century as the square of the number of centuries from a given 
epoch. This remarkable fact was discovered by Dr. Halley.* In a 
lunar eclipse the moon's longitude diners from that of the sun, at the 
middle of the eclipse, by exactly 180° ; and since the sun's lon- 
gitude at any given time of the year is known, if we can learn 
the day and hour when an eclipse occurs, we shall of course know 
the longitude of the sun and moon. Now in the year 721 before 
the Christian era, on a specified day and hour, Ptolemy records a 
lunar eclipse to have happened, and to have been observed by 
the Chaldeans. The moon's longitude, therefore, for that time is 
known ; and as we know the mean motions of the moon at pre- 
sent, starting from that epoch, and computing, as may easily be 
done, the place which the moon ought to occupy at present at any 
given time, she is found to be actually nearly a degree and a half 
in advance of that place. Moreover, the same conclusion is 
derived from a comparison of the Chaldean observations with those 
made by an Arabian astronomer of the tenth century. 

This phenomenon at first led astronomers to apprehend that the 
moon encountered a resisting medium, which, by destroying at 
every revolution a small portion of her projectile force, would 
have the effect to bring her nearer and nearer to the earth and 
thus to augment her velocity. But in 1786, La Place demon- 
strated that this acceleration is one of the legitimate effects of the 
sun's disturbing force, and is so connected with changes in the 
eccentricity of the earth's orbit, that the moon will continue to be 
accelerated while that eccentricity diminishes, but when the eccen- 
tricity has reached its minimum (as it will do after many ages) 
and begins to increase, then the moon's motion will begin to be 
retarded, and thus her mean motions will oscillate forever about a 
mean value. 

244. The lunar inequalities which have been considered are such 

* Astronomer Royal of Great Britain, and cotemporary with Sir Isaac Newton. 



ECLIPSES. 143 

only as affect the moon's longitude ; but the sun's disturbing force 
also causes inequalities in the moon's latitude and parallax. Those 
of latitude alone require no less than twelve equations. Since 
the moon revolves in an orbit inclined to the ecliptic, it is easy to 
see that the oblique action of the sun must admit of a resolution 
into two forces, one of which being perpendicular to the moon's 
orbit, must effect changes in her latitude. Since also several of the 
inequalities already noticed involve changes in the length of the 
radius vector, it is obvious that the moon's parallax must be sub- 
ject to corresponding perturbations. 



CHAPTER VII. 



ECLIPSES. 



245. An eclipse of the moon happens, when the moon in its 
revolution about the earth, falls into the earth's shadow. An 
eclipse of the sun happens, when the moon, coming between the 
earth and the sun, covers either a part or the whole of the solar 
disk. An eclipse of the sun can occur only at the time of con- 
junction, or new moon ; and an eclipse of the moon, only at the 
time of opposition, or full moon. Were the moon's orbit in the 
same plane with that of the earth, or did it coincide with the 
ecliptic, then an eclipse of the sun would take place at every 
conjunction, and an eclipse of the moon at every opposition ; for 
as the sun and earth both lie in the ecliptic, the shadow of the 
earth must also extend in the same plane, being of course always 
directly opposite to the sun ; and since, as we shall soon see, the 
length of this shadow is much greater than the distance of the 
moon from the earth, the moon, if it revolved in the plane of the 
ecliptic, must pass through the shadow at every full moon. For 
similar reasons, the moon would occasion an eclipse of the sun, 
partial or total, in some portions of the earth at every new moon. 
But the lunar orbit is inclined to the ecliptic about 5°, so that the 
center of the moon, when she is furthest from her node, is 5° from 



144 THE MOON. 

the axis of the earth's shadow (which is always in the ecliptic ;) 
and, as we shall show presently, the greatest distance to which the 
shadow extends on each side of the ecliptic, that is, the greatest 
semi-diameter of the shadow, where the moon passes through it, 
is only about f of a degree, while the semi-diameter of the moon's 
disk is only about I of a degree ; hence the two semi-diame- 
ters, namely, that of the moon and the earth's shadow, cannot 
overlap one another, unless, at the time of new or full moon, the 
sun is at or very near the moon's node. In the course of the sun's 
apparent revolution around the earth once a year, he is succes- 
sively in every part of the ecliptic ; consequently, the conjunctions 
and oppositions of the sun and moon may occur at any part of the 
ecliptic, either when the sun is at the moon's node, (or when he 
is passing that point of the celestial vault on which the moon's 
node is projected as seen from the earth ;) or they may occur 
when the sun is 90° from the moon's node, where the lunar and 
solar orbits are at the greatest distance from each other ; or, finally, 
they may occur at any intermediate point. Now the sun, in his 
annual revolution, passes each of the moon's nodes on opposite 
sides of the ecliptic, and of course at opposite seasons of the 
year ; so that, for any given year, the eclipses commonly happen 
in two opposite months, as January and July, February and 
August, May and November. These, therefore, are called Node 
Months. 

246. If the sun were of the same size with the earth, the shadow 
of the earth would be cylindrical and infinite in length, since the 
tangents drawn from the sun to the earth (which form the bounda- 
ries of the shadow) would be parallel to each other ; but as the 
sun is a vastly larger body than the earth, the tangents converge 
and meet in a point at some distance behind the earth, forming a 
cone of which the earth is the base, and whose vertex (and of 
course its axis) lies in the ecliptic. A little reflection will also 
show us, that the form and dimensions of the shadow must be 
affected by several circumstances ; that the shadow must be of 
the greatest length and breadth when the sun is furthest from the 
earth ; that its figure will be slightly modified by the spheroidal 
figure of the earth ; and that the moon, being, at the time of its 



ECLIPSES. 145 

opposition, sometimes nearer to the earth, and sometimes further 
from it, will accordingly traverse it at points where its breadth 
varies more or less. 

247. The semi-angle of the cone of the earth's shadow, is equal 
to the sun's apparent semi-diameter, minus his horizontal par- 
allax. 

Let AS (Fig. 51,) be the semi-diameter of the sun, BE that of 
the earth, and EC the axis of the earth's shadow. Then the 
semi-angle of the cone of the earth's shadow ECB=AES — EAB, 

Fig. 51, 

A 



of which AES is the sun's semi-diameter and EAB his horizontal 
parallax ; and as both these quantities are known, hence the angle 
at the vertex of the shadow becomes known. Putting <5 for the 
the sun's semi-diameter, andp for his horizontal parallax, we have 
the semi-angle of the earth's shadow ECB=<5— p. 

248. At the mean distance of the earth from the sun, the length 
of the earth's shadow is about 860,000 miles, or more than three times 
the distance of the moon from the earth. 

In the right angled triangle ECB, right angled at B, the angle 
ECB being known, and the side EB, we can find the side EC. 

FB 

For sin. (S-p) : EB : : R : EC= t-= :• This value will vary 

v r/ sin. (o— p) 

with the sun's semi-diameter, being greater as that is less. Its 

mean value being 16' 1".5 and the sun's horizontal parallax being 

8".6, h—p=\& 52".9, and EB=3956.2. Hence, 

Sin. 15' 53" : Rad. : : 3956.2 : 856,275. 

Since the distance of the moon from the earth is 238,545 miles, 

the shadow extends about 3.6 times as far as the moon, and con- 

19 



146 THE MOON. 

sequently, the moon passes the shadow towards its broadest part, 
where its breadth is much more than sufficient to cover the moon's 
disk. 

249. TJie average breadth of the earth's shadow where it eclipses 
the moon is almost three times the moon's diameter. 

. Let mm' (Fig. 51,) represent a section of the earth's shadow 
where the moon passes through it, M being the center of the cir- 
cular section. Then the angle MEm will be the angular breadth 
of half the shadow. But, 

ME/?z=BmE — BCE ; that is, since BmE is the moon's horizon- 
tal parallax, (Art. 82,) and BCE equals the sun's semi-diameter 
minus his horizontal parallax ($—p,) therefore, putting P for the 
moon's horizontal parallax, we have 

MEm = Y-(5—p)= 1 P+p — 8; that is, since P = 57' 1" and 
8—p=l5> 52".9, therefore, 57' l"— 15' 52".9=41' 8".l, which is 
nearly three times 15' 33", the semi-diameter of the moon. Thus, 
it is seen how, by the aid of geometry, we learn to estimate vari- 
ous particulars respecting the earth's shadow, by means of simple 
data derived from observation. 

250. The distance of the moon from her node when she just 
touches the shadow of the earth, in a lunar eclipse, is called the 
Lunar Ecliptic Limit ; and her distance from the node in a solar 
eclipse, when the moon just touches the solar disk, is called the 
Solar Ecliptic Limit. The Limits are respectively the furthest 
possible distances from the node at which eclipses can take place. 

251. Tlie Lunar Ecliptic Limit is nearly 12 degrees. 

Let CN (Fig. 52,) be the sun's path, MN the moon's, and N the 
node. Let Ca be the semi-diameter of the earth's shadow, and 
Ma the semi-diameter of the moon. Since Ca and Ma are known 

Fig. 52. 




ECLIPSES. 147 

quantities, their sum CM is also known. The angle at N is 
known, being the inclination of the lunar orbit to the ecliptic. 
Hence, in the spherical triangle MCN, right angled at M,* by 
Napier's theorem, (Art. 132, Note,) 

Rad.xsin. CM=sin. CNxsin. MNC. 
The greatest apparent semi-diameter of the earth's shadow 
where the moon crosses it, computed by article 249, is 45' 52", 
and the moons greatest apparent semi-diameter, is 16' 45".5, 
which together, give MC equal to 62' 37". 5. Taking the incli- 
nation of the moon's orbit, or the angle MNC (what it generally 
is in these circumstances) at 5° 17', and we have Rad.xsin. 

62' 37".5=sin. CNxsin. 5° 17', or sin. CN= Rad Xsin> 62 ' 37 "- 5 , 

sin. 5° 17' 

and CN=1 1° 25' 40 ; '.t This is the greatest distance of the moon from 
her node at which an eclipse of the moon can take place. By 
varying the value of CM, corresponding to variations in the dis- 
tances of the sun and moon from the earth, it is found that if NC 
is less than 9°, there must be an eclipse ; but between this and the 
limit, the case is doubtful. 

When the moon's disk only comes in contact with the earth's 
shadow, as in figure 52', the phenomenon is called an appulse, 
when only a part of the disk enters the shadow, the eclipse is 
said to be partial, and total if the whole of the disk enters the 
the shadow. The eclipse is called central when the moon's center 
coincides with the axis of the shadow, which happens when the 
moon at the time of the eclipse is exactly at her node. 

252. Before the moon enters the earth's shadow, the earth be- 
gins to intercept from it portions of the sun's light, gradually in- 
creasing until the moon reaches the shadow. This partial light is 
called the moon's Penumbra. Its limits are ascertained by drawing 
the tangents AC'B' and A'C'B. (Fig. 51.) Throughout the space 
included between these tangents more or less of the sun's light is 
intercepted from the moon by the interposition of the earth : for 



* The line CM is to be regarded as the projection of the line which connects the 
centers of the moon and section of the earth's shadow, as seen from the earth, 
t Woodhouse's Astronomy, p. 718, 



148 THE MOON. 

it is evident, that, as the moon moves towards the shadow, she 
would gradually lose the view of the sun, until, on entering the 
shadow, the sun would be entirely hidden from her. 

253. The semi-angle of the Penumbra equals the sun's semi- 
diameter and horizontal parallax, or S+p. 

The angle hC'M (Fig. 51,)=AC'S=AES+B'AE. But AES is 
the sun's semi-diameter, and B AE is the sun's horizontal parallax, 
both of which quantities are known. 

254. The semi-angle of a section of the Penumbra, where the 
moon crosses it, equals the moon's horizontal parallax, plus the sun's, 
plus the sun's semi-diameter. 

The angle hEM (Fig. 51,) =EhC'+EC'h. But EhC'=P, the 
moon's horizontal parallax, and EC'H=5+p (Art. 252,) .*. hEM. 
=P+p+<5, all which are likewise known quantities. 

Hence, by means of these few elements, which are known from 
observation, we ascend to a complete knowledge of all the par- 
ticulars necessary to be known respecting the moon's penumbra. 

255. In the preceding investigations, we have supposed that 
the cone of the earth's shadow is formed by lines drawn from the 
sun, and touching the earth's surface. But the apparent diameter 
of the shadow is found by observation to be somewhat greater than 
would result from this hypothesis. The fact is accounted for by 
supposing that a portion of the solar rays which graze the earth's 
surface are absorbed and extinguished by the lower strata of the 
atmosphere. This amounts to the same thing as though the earth 
were larger than it is, in which case the moon's horizontal parallax 
would be increased ; and accordingly, in order that theory and 
observation may coincide, it is found necessary to increase the 
parallax by eV- 

256. In a total eclipse of the moon, its disk is still visible, 
shining with a dull red light. This light cannot be derived di- 
rectly from the sun, since the view of the sun is completely hid- 
den from the moon ; nor by reflexion from the earth, since the 
illuminated side of the earth is wholly turned from the moon ; but 



ECLIPSES. 149 

it is owing to refraction by the earth's atmosphere, by which a few 
scattered rays of the sun are bent round into the earth's shadow 
and conveyed to the moon, sufficient in number to afford the feeble 
light in question. £ 

257. In calculating an eclipse of the moon, we first learn from 
the tables in what month the sun, at the time of full moon in that 
month, is near the moon's node, or within the lunar ecliptic limit. 
This it must evidently be easy to determine, since the tables ena- 
ble us to find both the longitudes of the nodes, and the longitudes 
of the sun and moon, for every day of the year. Consequently, 
we can find when the sun has nearly the same longitude as one of 
the nodes, and also the precise moment when the longitude* of the 
moon is 180° from that of the sun, for this is the time of opposition, 
from which may be derived the time of the middle of the eclipse. 
Having the time of the middle of the eclipse, and the breadth 
of the shadow, (Art. 249.) and knowing, from the tables, the rate 
at which the moon moves per hour faster than the shadow, we can 
find how long it will take her to traverse half the breadth of the 
shadow ; and this time subtracted from the time of the middle 
of the eclipse, will give the beginning, and added to the time of 
the middle will give the end of the eclipse. Or if instead of the 
breadth of the shadow, we employ the breadth of the penumbra 
(Art. 253,) we may find, in the same manner, when the moon 
enters and when she leaves the penumbra. We see, therefore, 
how by having a few things known by observation, such as the 
sun and moon's semi-diameters, and their horizontal parallaxes, 
we rise, by the aid of trigonometry, to the knowledge of various 
particulars respecting the length and breadth of the shadow and 
of the penumbra. These being known, we next have recourse to 
the tables which contain all the necessary particulars respecting 
the motions of the sun and moon, together with equations or cor- 
rections, to be applied for all their irregularities. Hence it is com- 
paratively an easy task to calculate with great accuracy an eclipse 
of the moon. 

258. Let us then see how we may find the exact tjme of the be- 
ginning, end, duration, and magnitude, of a lunar eclipse. 



150 THE MOON. 



Let NG (Fig 53,) be the ecliptic, and Nag the moon's orbit, the 
sun being in A* when the moon is in opposition at a ; let N be 
the ascending node, and Aa the moon's latitude at the instant 



Fie. 53. 



of opposition. An hour afterwards the sun will have passed to 
A', and the moon to g, when the difference of longitude of the two 
bodies will be GA'. Then gli is the moon's hourly motion in lati- 
tude, and ah her hourly motion in longitude. As the character 
and form of the eclipse will depend solely upon the distances 
between the centers of the sun and moon, that is, upon the line 
gA', instead of considering the two bodies as both in motion, 
we may suppose the sun at rest in A, and the moon as advancing 
with a motion equal to the difference between its rate and that 
of the sun, a supposition which will simplify the calculation. 
Therefore, draw gd parallel and equal to A'A, join dA, and this 
line being equal to gA', the two bodies will be in the same relative 
situation as if the sun were at A' and the moon at g. Join da and 
produce the line da both ways, cutting the ecliptic in F ; then 
daF will be the moon's Relative Orbit. Hence ai=ah— A A' =the 
difference of the hourly motions of the sun and moon, that is, the 
moon's relative motion in longitude, and ^i=the moon's hourly 
motion in latitude. 

Draw CD (Fig. 54,) to represent the ecliptic, and let A be the 
place of the sun. As the tables give the computation of the 
moon's latitude at every instant, consequently, we may take from 
them the latitude corresponding to the instant of opposition, and 
to one hour later ; and we may take also the sun's and moon's 
hourly motions in longitude. Take AD, AB, each equal to the 
relative motion, and Aa^the latitude in opposition, Dd=the lati- 

* It will be remarked that the point A really represents the center of the earth's 
shadow ; but as the real motions of the shadow are the same with the assumed motions 
iof the sun, the latter are used in conformity with the language of the tables. 



ECLIPSES. 

Fig. 54. 



151 




-0, 



tude one hour afterwards ; join da and produce the line da both 
ways, and it will represent the moon's relative orbit. Draw Bb 
at right angles to CD, and it will be the latitude an hour before 
opposition. At the time of the eclipse, the apparent distance of 
the center of the shadow from the moon is very small ; conse- 
quently, CD, cd, Dd, &c. may be regarded as straight lines. 
During the short interval between the beginning and end of an 
eclipse, the motion of the sun, and consequently that of the cen- 
ter of the shadow, may likewise be regarded as uniform. 

259. The various particulars that enter into the calculation of 
an eclipse are called its Elements ; and as our object is here merely 
to explain the method of calculating an eclipse of the moon, (refer- 
ring to the Supplement for the actual computation,) we may take 
the elements at their mean value. Thus, we will consider cd as 
inclined to CD 5° 9', the moon's horizontal parallax as 58', its semi- 
diameter as 16', and that of the earth's shadow as 42'. The line 
Am perpendicular to cd gives the point m for the place of the 
moon at the middle of the eclipse, for this line bisects the chord, 
which represents the path of the moon through the shadow ; and 
mM, perpendicular to CD, gives AM for the time of the middle 
of the eclipse before opposition, the number of minutes before op- 
position being the sarfle part of an hour that AM is of AB.* From 
the center A, with a radius equal to that of the earth's shadow 
(42') describe the semi-circle BLF, and it will represent the pro- 
jection of the shadow traversed by the moon. With a radius 
equal to the semi-diameter of the shadow and that of the moon 



* The situation of the moon when at m is called orbit opposition ; and her situation 
when at a, ecliptic opposition. 



152 THE MOON. 

(=42 / +16 / =58 / ) and with the center A, mark the two points c and 
/ on the relative orbit, and they will be the places of the center 
of the moon at the beginning and end of the eclipse. The per- 
pendiculars cC,fF, give the times AC and AF of the commence- 
ment and the end of the eclipse, and CM, or MF gives half the 
duration. From the centers c and f with a radius equal to the 
semi-diameter of the moon (16') describe circles, and they will 
each touch the shadow, (Euc. 3.12.) indicating the position of the 
moon at the beginning and end of the eclipse. If the same circle 
described from m is wholly within the shadow, the eclipse will be 
total ; if it is only partly within the shadow, the eclipse will be 
partial. With the center A, and radius equal to the semi-diame- 
ter of the shadow minus that of the moon (42' — 16' =26') mark 
the two points c',f, which will give the places of the center of the 
moon, at the beginning and end of total darkness, and MC, MF 7 
will give the corresponding times before and after the middle of 
the eclipse. Their sum will be the duration of total darkness. 

260. If the foregoing projection be accurately made from a scale, 
the required particulars of the eclipse may be ascertained by 
measuring on the same scale, the lines which respectively repre- 
sent them ; and we should thus obtain a near approximation to the 
elements of the eclipse. A more accurate determination of these 
elements may, however, be obtained by actual calculation. The 
general principles of the calculation will be readily understood. 

First, knowing ai, (Fig. 53,) the moon's relative longitude, and 
di, her latitude, we find the angle dai, which is the inclination of 
the moon's relative orbit. But dai=aAm ; and, in the triangle 
aAm, we have the angle at A, and the side Aa, being the moon's 
latitude at the time of opposition, which is given by the tables. 
Hence we can find the side Am. In the triangle AmM, (Fig. 54,) 
having the side Am and the angle AmM (— aAm) we can find AM 
= the arc of relative longitude described by the moon from the 
time of the middle of the eclipse to the time of opposition ; and 
knowing the moon's hourly motion in longitude, we can convert 
AM into time, and this subtracted from the time of opposition 
gives us the time of the middle of the eclipse. 



ECLIPSES. 153 

Secondly, since we know the length of the line Ac # (Fig. 54) 
and can easily find the angle cAC, we can thus obtain the side 
AC ; and AC — AM=MC, which arc, converted into time by com- 
paring it with the moon's hourly motion in longitude, gives us, 
when subtracted from the time of the middle of the eclipse, the 
time of the beginning of the eclipse, or when added to that of the 
middle, the time of the end of the eclipse. The sum of the two 
equals the whole duration. 

Thirdly, by a similar method we calculate the value of MC, 
which converted into time, and subtracted from the time of the 
middle of the eclipse, gives the commencement of total darkness, or 
when added gives the end of total darkness. Their sum is the 
duration of total darkness. 

Fourthly, the quantity of the eclipse is determined by supposing 
the diameter of the moon divided into twelve equal parts called 
Digits, and finding how many such parts lie within the shadow, 
at the time when the centers of the moon and the shadow are 
nearest to each other. Even when the moon lies wholly within 
the shadow, the quantity of the eclipse is still expressed by the num- 
ber of digits contained in that part of the line which joins the cen- 
ter of the shadow and the center of the moon, which is intercepted 
between the edge of the shadow and the inner edge of the moon. 

no 
Thus in figure 54, the number of digits eclipsed, equals — — - 

Ao—An Ao—(Am—nm) . . . , , 

= = i - — , an expression containing only known 

T \nl T \nl 

quantities. 



261. The foregoing will serve as an explanation of the general 
principles, on which proceeds the calculation of a lunar eclipse. 
The actual methods practiced employ many expedients to facili- 
tate the process, and to insure the greatest possible accuracy, the 
nature of which are explained and exemplified in Mason's Supple- 
ment to this work. 

262. The leading particulars respecting an Eclipse of the 
Sun, are ascertained very nearly like those of a lunar eclipse. The 

* This line is not represented in the figure, but may be easily imagined. 

20 



154 THE MOON. 

shadow of the moon travels over a portion of the earth, as the 
shadow of a small cloud, seen from an eminence in a clear day, 
rides along over hills and plains. Let us imagine ourselves stand- 
ing on the moon ; then we shall see the earth partially eclipsed by 
the shadow of the moon, in the same manner as we now see the 
moon eclipsed by the earth's shadow ; and we might proceed to 
find the length of the shadow, its breadth where it eclipses the 
earth, the breadth of the penumbra, and its duration and quantity, 
in the same way as we have ascertained these particulars for an 
eclipse of the moon. 

But, although the general characters of a solar eclipse might be 
investigated on these principles, so far as respects the earth at 
large, yet as the appearances of the same eclipse of the sun are 
very different at different places on the earth's surface, it is neces- 
sary to calculate its peculiar aspects for each place separately, a 
circumstance which makes the calculation of a solar eclipse much 
more complicated and tedious than of an eclipse of the moon. 
The moon, when she enters the shadow of the earth, is deprived 
of the light of the part immersed, and that part appears black 
alike to all places where the moon is above the horizon. But it is 
not so with a solar eclipse. We do not see this by the shadow 
cast on the earth, as we should do if we stood on the moon, but 
by the interposition of the moon between us and the sun ; and the 
sun may be hidden from one observer while he is in full view of 
another only a few miles distant. Thus, a small insulated cloud 
sailing in a clear sky, will, for a few moments, hide the sun from 
us, and from a certain space near us, while all the region around 
is illuminated. 

263. We have compared the motion of the moon's shadow over 
the surface of the earth to that of a cloud ; but its velocity is in- 
comparably greater. The mean motion of the moon around the 
earth is about 33' per hour ; but 33' of the moon's orbit is 2280 
miles, and the shadow moves of course at the same rate, or 2280 
miles per hour, traversing the entire disk of the earth in less than 
four hours. This is the velocity of the shadow when it passes 
perpendicularly over the earth ; when the direction of the axis of 
the shadow is oblique to the earth's surface, the velocity is increased 



ECLIPSES. 155 

in proportion of radius to the sine of obliquity. Thus the shadows 
of evening have a far more rapid motion than those of noon-day. 
Let us endeavor to form a just conception of the manner in 
which these three bodies, the sun, the earth, and the moon, are 
situated with respect to each other at the time of a solar eclipse. 
First, suppose the conjunction to take place at the node. Then 
the straight line which connects the centers of the sun and the 
earth, also passes through the center of the moon, and coincides 
with the axis of its shadow ; and, since the earth is bisected by 
the plane of the ecliptic, the shadow would traverse the earth in 
the direction of the terrestrial ecliptic, from west to east, passing 
over the middle regions of the earth. Here the diurnal motion of 
the earth being in the same direction with the shadow, but with a 
less velocity, the shadow will appear to move with a speed equal 
only to the difference between the two. Secondly, suppose the 
moon is on the north side of the ecliptic at the time of conjunction, 
and moving towards her descending node, and that the conjunc- 
tion takes place just within the solar ecliptic limit, say 16° from the 
node. The shadow will now not fall in the plane of the ecliptic, 
but a little northward of it, so as just to graze the earth near the 
pole of the ecliptic. The nearer the conjunction comes to the 
node, the further the shadow will fall from the pole of the ecliptic 
towards the equatorial regions. In certain cases, the shadow 
strikes beyond the pole of the earth ; and then its easterly motion 
being opposite to the diurnal motion of the places which it traver- 
ses, consequently its velocity is greatly increased, being equal to 
the sum of both. 

264. After these general considerations, we will now examine 
more particularly the method of investigating the elements of a 
solar eclipse. 

The length of the moon's shadow, is the first object of inquiry. 
The moon, as well as the earth, is at different distances from the 
sun at different times, and hence the length of her shadow varies, 
being always greatest when she is furthest from the sun. Also, 
since her distance from the earth varies, the section of the moon's 
shadow made by the earth, is greater in proportion as the moon is 



156 THE MOON. 

nearer the earth. The greatest eclipses of the sun, therefore, 
happen when the sun is in apogee,* and the moon in perigee. 

265. When the moon is at her mean distance from the earth, and 
from the sun, her shadow nearly reaches the earth's surface. 

Let S (Fig. 55,) represent the sun, D the moon, and T the 
earth. Then, the semi-angle of the cone of the moon's shadow, 
DKR, will, as in the case of the earth, (Art. 247,) equal SDR— 
DRK, of which SDR is the sun's apparent semi-diameter, as seen 
from the moon, and DRK, is the sun's horizontal parallax at the 
moon. Since, on account of the great distance of the sun, corn- 
Fig. 55. 



G. 



A 




pared with that of the moon, the semi-diameter of the sun as seen 
from the moon, must evidently be very nearly the same as 
when seen from the earth, and since on account of the minute- 
ness of the moon's semi-diameter when seen from the sun, the 
sun's horizontal parallax at the moon must be very small, we might, 
without much error, take the sun's apparent semi-diameter from 
the earth, as equal to the semi-angle of the cone of the moon's 
shadow ; but, for the sake of greater accuracy, let us estimate the 
value of the sun's semi-diameter and horizontal parallax at the 
moon. 

Now, SDR : STR : : ST : SDf : : 400 : 399 ; hence SDR = 

— STR=L0025 STR ; and the sun's mean semi-diameter STR 
399 

being 16.025, hence SDR=1.0025xl6.025=16.065=16' 3".9. ' 
Again, since parallax is inversely as the distance, the sun's hor- 
izontal parallax at the moon, is on account of her being nearer the 
sun ¥ ^ greater than at the earth ; but on account of her inferior 

* The sun is said to be in apogee, when the earth is in aphelion, 
t The apparent magnitude of an object being reciprocally as its distance from the 
eye. See Note, p. 86. 



ECLIPSES. 157 

size it is |fif less than at the earth. Hence, increasing the sun's 
horizontal parallax at the earth by the former fraction, and dimin- 
ishing it by the latter, we have -—x—— x 9" =2''. 5= the sun's 

oy y / y l £ 

horizontal parallax at the moon. Therefore, the semi-angle of the 
cone of the moon's shadow, which, as appears above, equals 
SDR— DRK, equals 16' 3".9 — 2".5=16' 1".4, which so nearly 
equals the sun's apparent semi-diameter, as seen from the earth, 
that we may adopt the latter as the value of the semi-angle of the 
shadow. Hence, sin. 16' 1".5 : 1080 (BD) : : Rad. : DK=231690. 
But the mean distance of the moon from the surface of the earth 
is 238545-3956=234589, which exceeds a little the mean length 
of the shadow as above. 

But when the moon is nearest the earth her distance from the 
center of the earth is only 221148 miles; and when the earth is 
furthest from the sun, the sun's apparent semi-diameter is only 
15' 45".5. By employing this number in the foregoing estimate, 
we shall find the length of the shadow 235630 miles ; and 
235630-221148=14482, the distance which the moon's shadow 
may reach beyond the center of the earth. 

266. The diameter of the moon's shadow where it traverses the 
earth, is, at its maximum, about 170 miles* 

In the triangle eTK, the angle at K=15' 45".5 (Art. 265,) the 
side Te=3956, and TK=14482. 

Or, 3956 : 14482 : : sin. 15' 45'.5 : sin. 57' 41".5. 

And 57' 41".5+15' 45".5=1° 13' 2T'=dTe, or the arc de. 

And 2de=2° 26' 54"=™. 

Hence 360 : 2.45 (=2° 26' 54") : : 24899f : 170 (nearly). 

267. The greatest portion of the earth's surface ever covered by 
the moon's j)enu?nbr a, is about 4393 miles. 

The semi-angle of the penumbra BID=BSD+SBR, of which 
BSD the sun's horizontal parallax at the moon =2".5, and SBR 
the sun's apparent semi-diameter =16' 3". 9, and hence BID is 

* This supposes the conjunction to take place at the node, and the shadow to strike 
the earth perpendicularly to its surface ; where it strikes obliquely, the section may be 
greater than this. 

t The equatorial circumference. 



158 



THE MOON, 



known* The moon's apparent semi-diameter BGD=16' 45".5. 
Therefore GDT is known, as likewise DT and TG. Hence the 
angle GTd may be found, and the arc dG and its double GH, 
which equals the angular breadth of the penumbra. It may be 
converted into miles by stating a proportion as in article 266. 
On making the calculation it will be found to be 4393 miles. 

268. The apparent diameter of the moon is sometimes larger 
than that of the sun, sometimes smaller, and sometimes exactly 
equal to it. Suppose an observer placed on the right line which 
joins the centers of the sun and moon ; if the apparent diameter of 
the moon is greater than that of the sun, the eclipse will be total. 
If the two diameters are equal, the moon's shadow just reaches the 
earth, and the sun is hidden but for a moment from the view of 
spectators situated in the line which the vertex of the shadow de- 
scribes on the surface of the earth. But if, as happens when the 
moon comes to her conjunction in that part of her orbit which is 
towards her apogee, the moon's diameter is less than the sun's, 
then the observer will see a ring of the sun encircle the moon, 
constituting an annular eclipse. (Fig. 55'.) 

Fig. 55'. 




269. Eclipses of the sun are modified by the elevation of the 
moon above the horizon, since its apparent diameter is augmented 



ECLIPSES. 159 

as its altitude is increased, (Art. 217.) This effect, combined with 
that of parallax, may so increase or diminish the apparent distance 
between the centers of the sun and moon, that from this cause 
alone, of two observers at a distance from each other, one might 
see an eclipse which was not visible to the other.* If the hori- 
zontal diameter of the moon differs but little from the apparent 
diameter of the sun, the case might occur where the eclipse would 
be annular over the places where it was observed morning and 
evening, but total where it was observed at mid-day. 

The earth in its diurnal revolution and the moon's shadow both 
move from west to east, but the shadow moves faster than the 
earth ; hence the moon overtakes the sun on its western limb and 
crosses it from west to east. The excess of the apparent diame- 
ter of the moon above that of the sun in a total eclipse is so small, 
that total darkness seldom continues longer than four minutes, and 
can never continue so long as eight minutes. An annular eclipse 
may last 12m. 24s. 

Since the sun's ecliptic limits are more than 17° and the moon's 
less than 12°, eclipses of the sun are more frequent than those of 
the moon. Yet lunar eclipses being visible to every part of the 
terrestrial hemisphere opposite to the sun, while those of the sun 
are visible only to the small portion of the hemisphere on which 
the moon's shadow falls, it happens that for any particular place 
on the earth, lunar eclipses are more frequently visible than solar. 
In any year, the number of eclipses of both luminaries cannot be 
less than two nor more than seven : the most usual number is four,, 
and it is very rare to have more than six. A total eclipse of the 
moon frequently happens at the next full moon after an eclipse of 
the sun. For since, in an eclipse of the sun, the sun is at or near 
one of the moon's nodes, the earth's shadow must be at or near 
the other node, and may not have passed so far from the node as 
the lunar ecliptic limits, before the moon overtakes it. 

270. It has been observed already, that were the spectator on 
the moon instead of on the earth, he would see the earth eclipsed 
by the moon, and the calculation of the eclipse would be very sim- 
ilar to that of a lunar eclipse ; but to an observer on the earth the 

* Biot, Ast. Phys. p. 401. 



160 THE MOON. 

eclipse does not of course begin when the earth first enters the 
moon's shadow, and it is necessary to determine not only what 
portion of the earth's surface will be covered by the moon's sha- 
dow, but likewise the path described by its center relative to va- 
rious places on the surface of the earth. This is known when the 
latitude and longitude of the center of the shadow on the earth, is 
determined for each instant. The latitude and longitude of the 
moon are found on the supposition that the spectator views it from 
the center of the earth, whereas his position on the surface changes, 
in consequence of parallax, both the latitude and longitude, and 
the amount of these changes must be accurately estimated, before 
the appearance of the eclipse at any particular place can be fully 
determined. 

The details of the method of calculating a solar eclipse cannot 
be understood in any way so well, as by actually performing the 
process according to a given example. For such details therefore 
the reader is referred to the Supplement. 

271. In total eclipses of the sun, there has sometimes been ob- 
served a remarkable radiation of light from the margin of the sun. 
This has been ascribed to an illumination of the solar atmosphere, 
but it is with more probability owing to the zodiacal light (Art. 
152,) which at that time is projected around the sun, and which is 
of such dimensions as to extend far beyond the solar orb. # 

A total eclipse of the sun is one of the most sublime and impres- 
sive phenomena of nature. Among barbarous tribes it is ever con- 
templated with fear and astonishment, while among cultivated na- 
tions it is recognized, from the exactness with which the time of 
occurrence and the various appearances answer to the prediction, 
as affording one of the proudest triumphs of astronomy. By 
astronomers themselves it is of course viewed with the highest 
interest, not only as verifying their calculations, but as contribu- 
ting to establish beyond all doubt the certainty of those grand 
laws, the truth of which is involved in the result. During the 
eclipse of June, 1806, which was one of the most remarkable on 

* See an excellent description and delineation of this appearance as it was exhibited 
in the eclipse of 1806, in the Transactions of the Albany Institute, by the late Chan- 
cellor De Witt. 



LONGITUDE. 161 

record, the time of total darkness, as seen by the author of this 
work, was about mid-day. The sky was entirely cloudless, but 
as the period of total obscuration approached, a gloom pervaded 
all nature. When the sun was wholly lost sight of, planets and 
stars came into view ; a fearful pall hung upon the sky, unlike 
both to night and to twilight ; and, the temperature of the air rap- 
idly declining, a sudden chill came over the earth. Even the ani- 
mal tribes exhibited tokens of fear and agitation. 

From 1831 to 1838 was a period remarkable for great eclipses 
of the sun, in which time there were no less than five of the most 
remarkable character. The next total eclipse of the sun, visible 
in the United States, will occur on the 7th of August, 1869. 



CHAPTER VIII 



LONGITUDE TIDES. 



272. As eclipses of the sun afford one of the most approved 
methods of finding the longitudes of places, our attention is natu- 
rally turned next towards that subject. 

The ancients studied astronomy in order that they might read 
their destinies in the stars : the moderns, that they may securely 
navigate the ocean. A large portion of the refined labors of 
modern astronomy, has been directed towards perfecting the as- 
tronomical tables with the view of finding the longitude at sea, — 
an object manifestly worthy of the highest efforts of science, con- 
sidering the vast amount of property and of human life involved 



in navigation. 



273. The difference of longitude between two places may be found 
by any method, by which we can ascertain the difference of their local 
times, at the same instant of absolute time. 

As the earth turns on its axis from west to east, any place that 
lies eastward of another will come sooner under the sun, or will 

21 



162 THE MOON. 

have the sun earlier on the meridian, and consequently, in respect 
to the hour of the day, will be in advance of the other at the 
rate of one hour for every 15°, or four minutes of time for each 
degree. Thus, to a place 15° east of Greenwich, it is 1 o'clock, 
P. M. when it is noon at Greenwich; and to a place 15° west of 
that meridian, it is 11 o'clock, A. M. at the same instant. Hence, 
the difference of time at any two places, indicates their difference 
of longitude. 

274. The easiest method of finding the longitude is by means 
of an accurate time piece, or chronometer. Let us set out from 
London with a chronometer accurately adjusted to Greenwich 
time, and travel eastward to a certain place, where the time is 
accurately kept, or may be ascertained by observation. We find, 
for example, that it is 1 o'clock by our chronometer, when it is 
2 o'clock and 30 minutes at the place of observation. Hence, 
the longitude is 15x1.5=22*° E. Had we travelled westward 
until our chronometer was an hour and a half in advance of the 
time at the place of observation, (that is, so much later in the 
day,) our longitude would have been 22£° W. But it would not 
be necessary to repair to London in order to set our chronometer 
to Greenwich time. This might be done at any observatory, or 
any place whose longitude had been accurately determined. For 
example, the time at New York is 4h. 56m. 4 9 .5 behind that of 
Greenwich. If, therefore, we set our chronometer so much be- 
fore the true time at New York, it will indicate the time at Green- 
wich. Moreover, on arriving at different places, any where on 
the earth, whose longitude is accurately known, we may learn 
whether our chronometer keeps accurate time or not, and if not, 
the amount of its error. Chronometers have been constructed of 
such an astonishing degree of accuracy, as to deviate but a few 
seconds in a voyage from London to Baffin's Bay and back, during 
an absence of several years. But chronometers which are suffi- 
ciently accurate to be depended on for long voyages, are too ex- 
pensive for general use, and the means of verifying their accuracy 
are not sufficiently easy. Moreover, chronometers by being trans- 
ported from one place to another, change their daily rate, or de- 
part from that mean rate which they preserve at a fixed station. 



LONGITUDE. 163 

A chronometer, therefore, cannot be relied on for determining the 
longitudes of places where the greatest degree of accuracy is re- 
quired, especially where the instrument is conveyed over land, 
although the uncertainty attendant on one instrument may be 
nearly obviated by employing several and taking their mean 
results.* 

275. Eclipses of the sun and moon are sometimes used for de- 
termining the longitude. The exact instant of immersion or of 
emersion, or any other definite moment of the eclipse which pre- 
sents itself to two distant observers, affords the means of com- 
paring their difference of time, and hence of determining their 
difference of longitude. Since the entrance of the moon into 
the earth's shadow, in a lunar eclipse, is seen at the same instant 
of absolute time at all places where the eclipse is visible, (Art. 
262,) this observation would be a very suitable one for finding 
the longitude were it not that, on account of the increasing dark- 
ness of the penumbra near the boundaries of the shadow, it is 
difficult to determine the precise instant when the moon enters the 
shadow. By taking observations on the immersions of known 
spots on the lunar disk, a mean result may be obtained which will 
give the longitude with tolerable accuracy. In an eclipse of the 
sun, the instants of immersion and emersion maybe observed with 
greater accuracy, although, since these do not take place at the 
same instant of absolute time, the calculation of the longitude from 
observations on a solar eclipse are complicated and laborious. 

A method very similar to the foregoing, by observations on 
eclipses of Jupiter's satellites, and on occultations of stars, will 
be mentioned hereafter. 

276. The Lunar method of finding the longitude, at sea, is in 
many respects preferable to every other. It consists in measuring 
(with a sextant) the angular distance between the moon and the 
sun, or between the moon and a star, and then turning to the Nau- 
tical Almanac,f and finding what time it was at Greenwich when 

* Woodhouse, p. 838. 

t The Nautical Almanac is a book published annually by the British Board of 
Longitude, containing various tables and astronomical information for the use of 



164 THE MOON. 

that distance was the same. The moon moves so rapidly, that this 
distance will not be the same except at very nearly the same in- 
stant of absolute time. For example, at 9 o'clock, A. M., at a cer- 
tain place, we find the angular distance of the moon and the sun to 
be 72° ; and on looking into the Nautical Almanac, we find that 
at the time when this distance was the same for the meridian of 
Greenwich was 1 o'clock, P. M. ; hence we infer that the longi- 
tude of the place is four hours, or 60° west. 

The Nautical Almanac contains the true angular distance of 
the moon from the sun, from the four large planets, (Venus, Mars, 
Jupiter, and Saturn,) and from nine bright fixed stars, for the be- 
ginning of every third hour of mean time for the meridian of 
Greenwich ; and the mean time corresponding to any intermediate 
hour, may be found by proportional parts.* 

277. It would be a very simple operation to determine the lon- 
gitude by Lunar Distances, if the process as described in the 
preceding article were all that is necessary ; but the various cir- 
cumstances of parallax, refraction, and dip of the horizon, would 
differ more or less at the two places, even were the bodies whose 
distances were taken in view from both, which is not necessarily 
the case. The observations, therefore, require to be reduced to 
the center of the earth, being cleared of the effects of parallax and 
refraction. Hence, three observers are necessary in order to take 
a lunar distance in the most exact manner, viz. two to measure 
the altitudes of the two bodies respectively, at the same time that 
the third takes the angular distance between them. The altitudes 
of the two luminaries at the time of observation must be known, 
in order to estimate the effects of parallax and refraction. 

278. Although the lunar method of finding the longitude at 
sea has many advantages over the other methods in use, yet it 



navigators. The American Almanac also contains a variety of astronomical informa- 
tion, peculiarly interesting to the people of the United States, in connexion with a 
vast amount of statistical matter. It is well deserving a place in the library of the 
student. 

* See Bow ditch's Navigator, Tenth Ed. p. 226. 



TIDES. 165 

has also its disadvantages. One is, the great exactness requisite 
in observing the distance of the moon from the sun or star, as a 
small error in the distance makes a considerable error in the longi- 
tude. The moon moves at the rate of about a degree in two 
hours, or one minute of space in two minutes of time. There- 
fore, if we make an error of one minute in observing the distance, 
we make an error of two minutes in time, or 30 miles of longitude 
at the equator. A single observation with the best sextants, may- 
be liable to an error of more than half a minute ; but the accuracy 
of the result may be much increased by a mean of several obser- 
vations taken to the east and west of the moon. The imperfection 
of lunar tables was until recently considered as an objection to this 
method. Until within a few years, the best lunar tables were 
frequently erroneous to the amount of one minute, occasioning an 
error of 30 miles. The error of the best tables now in use will 
rarely exceed 7 or 8 seconds.* 

TIDES. 

279. The tides are an alternate rising and falling of the waters 
of the ocean, at regular intervals. They have a maximum and a 
minimum twice a day, twice a month, and twice a year. Of the 
daily tide, the maximum is called High tide, and the minimum 
Low tide. The maximum for the month is called Spj~ing tide, and 
the minimum Neap tide. The rising of the tide is called Flood 
and its falling Ebb tide. 

Similar tides, whether high or low, occur on opposite sides of 
the earth at once. Thus at the same time it is high tide at any 
given place, it is also high tide on the inferior meridian, and the 
same is true of the low tides. 

The interval between two successive high tides is 12h. 25m. ; 
or, if the same tide be considered as returning to the meridian, 
after having gone around the globe, its return is about 50 minutes 
later than it occurred on the preceding day. In this respect, as 
well as in various others, it corresponds very nearly to the motions 
of the moon. 

* Brinkley's Elements of Astronomy, p. 241 



166 THE MOON. 

The average height for the whole globe is about 2£ feet ; or, 
if the earth were covered uniformly with a stratum of water, the 
difference between the two diameters of the oval would be 5 feet, 
©r more exactly 5 feet and 8 inches ; but its natural height at 
various places is very various, sometimes rising to 60 or 70 feet, 
and sometimes being scarcely perceptible. At the same place 
also the phenomena of the tides are very different at different 
times. 

Inland lakes and seas, even those of the largest class, as Lake 
Superior, or the Caspian, have no perceptible tide. 

280. Tides are caused by the unequal attraction of the sun and 
moon upon different parts of the earth. 

Suppose the projectile force by which the earth is carried for- 
ward in her orbit, to be suspended, and the earth to fall towards 
one of these bodies, the moon, for example, in consequence of 
their mutual attraction. Then, if all parts of the earth fell 
equally towards the moon, no derangement of its different parts 
would result, any more than of the particles of a drop of water 
in its descent to the ground. But if one part fell faster than an- 
other, the different portions would evidently be separated from 
each other. Now this is precisely what takes place with respect 
to the earth in its fall towards the moon. The portions of the 
earth in the hemisphere next to the moon, on account of being 
nearer to the center of attraction, fall faster than those in the op- 
posite hemisphere, and consequently leave them behind. The 
solid earth, on account of its cohesion, cannot obey this impulse, 
since all its different portions constitute one mass, which is acted 
on in the same manner as though it were all collected in the cen- 
ter ; but the waters on the surface, moving freely under this im- 
pulse, endeavor to desert the solid mass and fall towards the 
moon. For a similar reason the waters in the opposite hemisphere 
falling less towards the moon than the solid earth, are left behind, 
or appear to rise from the center of the earth. 

281. Let DEFG (Fig. 56,) represent the globe ; and, for the sake 
of illustrating the principle, we will suppose the waters entirely to 
cover the globe at a uniform depth. Let defg represent the solid 



TIDES. 



167 




globe, and the circular ring exterior to 
it, the covering of waters. Let C be 
the center of gravity of the solid mass, 
A that of the hemisphere next to the 
moon, and B that of the remoter hemi- 
sphere. Now the force of attraction 
exerted by the moon, acts in the same 
manner as though the solid mass were 
all concentrated in C, and the waters 
of each hemisphere at A and B respec- 
tively ; and (the moon being supposed above E) it is evident that 
A will tend to leave C, and C to leave B behind. The same must 
evidently be true of the respective portions of matter, of which 
these points are the centers of gravity. The waters of the globe 
will thus be reduced to an oval shape, being elongated in the direc- 
tion of that meridian which is under the moon, and flattened in 
the intermediate parts, and most of all at points ninety degrees dis- 
tant from that meridian. 

Were it not, therefore, for impediments which prevent the force 
from producing its full effects, we might expect to see the great 
tide-wave, as the elevated crest is called, always directly beneath 
the moon, attending it regularly around the globe. But the in- 
ertia of the waters prevents their instantly obeying the moon's 
attraction, and the friction of the waters on the bottom of the 
ocean, still further retards its progress. It is not therefore until 
several hours (differing at different places) after the moon has 
passed the meridian of a place, that it is high tide at that place. 



282. The sun has a similar action to the moon, but only one 
third as great. On account of the great mass of the sun com- 
pared with that of the moon, we might suppose that his action 
in raising the tides would be greater than the moon's ; but the 
nearness of the moon to the earth more than compensates for 
the sun's greater quantity of matter. Let us, however, form a 
just conception of the advantage which the moon derives from her 
proximity. It is not that her actual amount of attraction is thus 
rendered greater than that of the sun ; but it is that her attraction 
for the different parts of the earth is very unequal, while that of 



168 THE MOON. 

the sun is nearly uniform. It is the inequality of this action, and 
not the absolute force, that produces the tides. The diameter of 
the earth is ^ of the distance of the moon, while it is less than 
ro oo o °f the distance of the sun. 

283. Having now learned the general cause of the tides, we 
will next attend to the explanation of particular phenomena. 

The Spring tides, or those which rise to an unusual height 
twice a month, are produced by the sun and moon's acting to- 
gether ; and the Neap tides, or those which are unusually low 
twice a month, are produced by the sun and moon's acting in 
opposition to each other. The Spring tides occur at the syzygies ; 
the Neap tides at the quadratures. At the time of new moon, 
the sun and moon both being on the same side of the earth, and 
acting upon it in the same line, their actions conspire, and the 
sun may be considered as adding so much to the force of the 
moon. We have already explained how the moon contributes to 
raise a tide on the opposite side of the earth. But the sun as well 
as the moon raises its own tide-wave, which, at new moon, coin- 
cides with the lunar tide-wave. At full moon, also, the two lumina- 
ries conspire in the same way to raise the tide ; for we must recol- 
lect that each body contributes to raise the tide on the opposite 
side of the earth as well as on the side nearest to it. At both the 
conjunctions and oppositions, therefore, that is, at the syzygies, 
we have unusually high tides. But here also the maximum effect 
is not at the moment of the syzygies, but 36 hours afterwards. 

At the quadratures, the solar wave is lowest where the lunar 
wave is highest ; hence the low tide produced by the sun is sub- 
tracted from high water and produces the Neap tides. Moreover, 
at the quadratures the solar wave is highest where the lunar wave 
is lowest, and hence is to be added to the height of low water at 
the time of Neap tides. Hence the difference between high and 
low water is only about half as great at Neap tide as at Spring tide. 

284. The power of the moon or of the sun to raise the tide is 
found by the doctrine of universal gravitation to be inversely as 
the cube of the distance.* The variations of distance in the sun are 

* La Place, Syst. du Monde, 1. iv, c. x. 



TIDES. 



169 



not great enough to influence the tides very materially, but the 
variations in the moon's distances have a striking effect. The 
tides which happen when the moon is in perigee, are considerably 
greater than when she is in apogee ; and if she happens to be in 
perigee at the time of the syzygies, the spring tide is unusually 
high. When this happens at the equinoxes, the highest tides of 
the year are produced. 

285. The declinations of the sun and moon have a considerable 
influence on the height of the tide. When the moon, for example, 
has no declination, or is in the equator, as in figure 57,* the rota- 
tion of the earth on its axis NS will make the two tides exactly 
equal on opposite sides of the earth. Thus a place which is car- 
ried through the parallel TT' will have the height of one tide T2 
and the other tide T'3. The tides are in this case greatest at the 
equator, and diminish gradually to the poles, where it will be low 
water during the whole day. When the moon is on the north side 
of the equator, as in figure 58, at her greatest northern declination, 
Fig. 57. Fig. 58. 





a place describing the parallel TT' will have T'3 for the height of 
the tide when the moon is on the superior meridian, and T2 for 
the height when the moon is on the inferior meridian. Therefore, 
all places north of the equator will have the highest tide when the 
moon is above the horizon, and the lowest when she is below it ; 
the difference of the tides diminishing towards the equator, where 



* Diagrams like these are apt to mislead the learner, by exhibiting the protuberance 
occasioned by the tides as much greater than the reality. We must recollect that it 
amounts, at the highest, to only a very few feet in eight thousand miles. Were the 
diagram, therefore, drawn in just proportions, the alterations of figure produced by the 
tides would be wholly insensible. 

22 



170 THE MOON. 

they are equal. In like manner, places south of the equator have 
the highest tides when the moon is below the horizon, and the 
lowest when she is above it. When the moon is at her greatest 
declination, the highest tides will take place towards the tropics. 
The circumstances are all reversed when the moon is south of the 
equator.* 

286. The motion of the tide- wave, it should be remarked, is not 
a progressive motion, but a mere undulation, and is to be carefully 
distinguished from the currents to which it gives rise. If the 
ocean completely covered the earth, the sun and moon being in the 
equator, the tide-wave would travel at the same rate as the earth 
on its axis. Indeed, the correct way of conceiving of the tide- 
wave, is to consider the moon at rest, and the earth in its rotation 
from west to east as bringing successive portions of water under 
the moon, which portions being elevated successively at the same 
rate as the earth revolves on its axis, have a relative motion west- 
ward in the same degree. 

287. The tides of rivers, narrow bays, and shores far from the 
main body of the ocean, are not produced in those places by the 
direct action of the sun and moon, but are subordinate waves 
propagated from the great tide-wave. 

Lines drawn through all the adjacent parts of any tract of wa- 
ter, which have high water at the same time, are called cotidal 
lines.-\ We may, for instance, draw a line through all places in 
the Atlantic Ocean which have high tide on a given day at 1 o'clock, 
and another through all places which have high tide at 2 o'clock. 
The cotidal line for any hour may be considered as representing 
the summit or ridge of the tide-wave at that time ; and could the 
spectator, detached from the earth, perceive the summit of the 
wave, he would see it travelling round the earth in the open ocean 
once in twenty four hours, followed by another twelve hours dis- 
tant, and both sending branches into rivers, bays, and other open- 
ings into the main land. These latter are called Derivative tides, 



* Edin. Encyc. Art. Astronomy, p. 623. 

t Whewell, Phil. Transaction for 1833, p. 148. 



TIDES. 171 

while those raised directly by the action of the sun and moon, are 
called Primitive tides. 

288. The velocity with which the wave moves will depend on 
various circumstances, but principally on the depth, and probably 
on the regularity of the channel. If the depth be nearly uniform, 
the cotidal lines will be nearly straight and parallel. But if some 
parts of the channel are deep while others are shallow, the tide 
will be detained by the greater friction of the shallow places, and 
the cotidal lines will be irregular. The direction also of the de- 
rivative tide, may be totally different from that of the primitive. 
Thus, (Fig. 59,) if the great tide- Fig. 59. 

wave, moving from east to west, 
be represented by the lines 1, 2, 
3, 4, the derivative tide which is 
propagated up a river or bay, 
will be represented by the cotidal 
lines 3, 4, 5, 6, 7. Advancing 
faster in the channel than next 
the banks, the tides will lag be- 
hind towards the shores, and the 
cotidal lines will take the form 
of curves as represented in the 
diagram. 

289. On account of the retarding influence of shoals, and an 
uneven, indented coast, the tide-wave travels more slowly along 
the shores of an island than in the neighboring sea, assuming con- 
vex figures at a little distance from the island and on opposite 
sides of it. These convex lines sometimes meet and become 
blended in such a manner as to create singular anomalies in a sea 
much broken by islands, as well as on coasts indented with numer- 
ous bays and rivers.* Peculiar phenomena are also produced, 
when the tide flows in at opposite extremities of a reef or island, 
as into the two opposite ends of Long Island Sound. In certain 



* See an excellent representation and description of these different phenomena by 
Professor Whewell, Phil. Trans. 1833, p. 153. 




172 THE MOON. 

cases a tide-wave is forced into a narrow arm of the sea, and 
produces very remarkable tides. The tides of the Bay of Fundy 
(the highest in the world) sometimes rise to the height of 60 or 70 
feet ; and the tides of the river Severn, near Bristol in England, 
rise to the height of 40 feet. 

290. The Unit of Altitude of any place, is the height of the 
maximum tide after the syzygies, (Art. 283,) being usually about 
36 hours after the new or full moon. But as the amount of this 
tide would be affected by the distance of the sun and moon from 
the earth, (Art. 284,) and by their declinations, (Art. 285,) these 
distances are taken at their mean value, and the luminaries are 
supposed to be in the equator ; the observations being so reduced 
as to conform to these circumstances. The unit of altitude can be 
ascertained by observation only. The actual rise of the tide de- 
pends much on the strength and direction of the wind. When 
high winds conspire with a high flood tide, as is frequently the 
case near the equinoxes, the tide rises to a very unusual height. 
We subjoin from the American Almanac a few examples of the 
unit of altitude for different places. 

Feet. 

Cumberland, head of the Bay of Fundy, 71 
Boston, . . . . . 11£ 

New Haven, .... 8 

New York, .... 5 

Charleston, S. C, . .6 

291. The Establishment of any port is the mean interval between 
noon and the time of high water, on the day of new or full moon. 
As the interval for any given place is always nearly the same, it 
becomes a criterion of the retardation of the tides at that place. 
On account of the importance to navigation of a correct know- 
ledge of the tides, the British Board of Admiralty, at the sugges- 
tion of the Royal Society, recently issued orders to their agents 
in various important naval stations, to have accurate observations 
made on the tides, with the view of ascertaining the establishment 
and various other particulars respecting each station;* and the 

* Lubbock, Report on the Tides, 1833. 



TIDES. 1 73 

government of the United States is prosecuting similar investiga- 
tions respecting our own ports. 

292. According to Professor Whewell,* the tides on the coast 
of North America are derived from the great tide-wave of the 
South Atlantic, which runs steadily northward along the coast to 
the mouth of the Bay of Fundy, where it meets the northern tide- 
wave flowing in the opposite direction. Hence he accounts for 
the high tides of the Bay of Fundy. 

293. The largest lakes and inland seas have no perceptible 
tides. This is asserted by all writers respecting the Caspian and 
Euxine, and the same is found to be true of the largest of the 
North American lakes, Lake Superior. f 

Although these several tracts of water appear large when taken 
by themselves, yet they occupy but small portions of the surface 
of the globe, as will appear evident from the delineation of them 
on an artificial globe. Now we must recollect that the primitive 
tides are produced by the unequal action of the sun and moon 
upon the different parts of the earth ; and that it is only at points 
whose distance from each other bears a considerable ratio to the 
whole distance of the sun or the moon, that the inequality of ac- 
tion becomes manifest. The space required is larger than either 
of these tracts of water. It is obvious also that they have no op- 
portunity to be subject to a derivative tide. 

294. To apply the theory of universal gravitation to all the va- 
rying circumstances that influence the tides, becomes a matter of 
such intricacy, that La Place pronounces " the problem of the 
tides" the most difficult problem of celestial mechanics. 

295. The Atmosphere that envelops the earth, must evidently be 
subject to the action of the same forces as the covering of waters, 
and hence we might expect a rise and fall of the barometer, indi- 
cating an atmospheric tide corresponding to the tide of the ocean. 



* Phil. Trans. 1833, p. 172. 

t See Experiments of Gov. Cass, Am. Jour. Science. 



174 THE PLANETS. 

La Place has calculated the amount of this aerial tide. It is too 
inconsiderable to be detected by changes in the barometer, unless 
by the most refined observations. Hence it is concluded, that the 
fluctuations produced by this cause are too slight to affect me- 
teorological phenomena in any appreciable degree.* 



CHAPTER IX. 

OF THE PLANETS THE INFERIOR PLANETS, MERCURY AND VENUS. 

296. The name planet signifies a wanderer,] and is applied to 
this class of bodies because they shift their positions in the heav- 
ens, whereas the fixed stars constantly maintain the same places 
with respect to each other. The planets known from a high an- 
tiquity, are Mercury, Venus, Earth, Mars, Jupiter, and Saturn. 
To these, in 1781, was added Uranus, J (or Herschel, as it is some- 
times called from the name of its discoverer,) and, as late as the 
commencement of the present century, four more were added, 
namely, Ceres, Pallas, Juno, and Vesta. These bodies are desig- 
nated by the following characters : 



1. Mercury 


$ 


7. Ceres 


? 


2. Venus 


9 


8. Pallas 


£ 


3. Earth 


e 


9. Jupiter 


U 


4. Mars 


$ 


10. Saturn 


fr 


5. Vesta 


ft 


11. Uranus 


W 


6. Juno 


$ 







The foregoing are called the primary planets. Several of these 
have one or more attendants, or satellites, which revolve around 
them, as they revolve around the sun. The earth has one satel- 
lite, namely, the moon ; Jupiter has four ; Saturn, seven ; and Ura- 

* Bowditch's La Place, II. 797. 
t From the Greek, n\awrv5' 
t From Cvpavos* 



DISTANCES PROM THE SUN. 175 

nus, six. These bodies also are planets, but in distinction from the 
others they are called secondary planets. Hence, the whole num- 
ber of planets are 29, viz. 11 primary, and 18 secondary planets. 

297. With the exception of the four new planets, these bodies 
have their orbits very nearly in the same plane, and are never seen 
far from the ecliptic. Mercury, whose orbit is most inclined of 
all, never departs further from the ecliptic than about 7°, while 
most of the other planets pursue very nearly the same path with 
the earth, in their annual revolution around the sun. The new 
planets, however, make wider excursions from the plane of the 
ecliptic, amounting, in the case of Pallas, to 34£°. 

298. Mercury and Venus are called inferior planets, because 
they have their orbits nearer to the sun than that of the earth ; 
while all the others, being more distant from the sun than the 
earth, are called superior planets. The planets present great di- 
versities among themselves in respect to distance from the sun, 
magnitude, time of revolution, and density. They differ also in 
regard to satellites, of which, as we have seen, three have respec- 
tively four, six, and seven, while more than half have none at all. 
It will aid the memory, and render our view of the planetary sys- 
tem more clear and comprehensive, if we classify, as far as possi- 
ble, the various particulars comprehended under the foregoing 
heads. 

299. DISTANCES FROM THE SUN.* 

1. Mercury, 37,000,000 0.3870981 

2. Venus, 68,000,000 0.7233316 

3. Earth, 95,000,000 1.0000000 

4. Mars, 142,000,000 1.5236923 

5. Vesta, 225,000,000 2.3678700 

6. Juno, \ 2.6690090 

7. Ceres, ( 261,000,000 2.7672450 

8. Pallas, ) 2.7728860 

* The distance in miles, as expressed in the first column, in round numbers, is to be 
treasured up in the memory, while the second column expresses the relative distances, 
that of the earth being 1, from which a more exact determination may be made, when 
required, the earth's distance being taken at 94,885,491. (Baily.) 



176 



THE PLANETS. 



9. Jupiter, 

10. Saturn, 

11. Uranus, 



485,000,000 

890,000,000 

1800,000,000 



5.2027760 

9.5387861 

19.1823900 



The dimensions of the planetary system are seen from this 
table to be vast, comprehending a circular space thirty-six hun- 
dred millions of miles in diameter. A railway car, travelling con- 
stantly at the rate of 20 miles an hour, would require more than 
20,000 years to cross the orbit of Uranus. 

It may aid the memory to remark, that in regard to the planets 
nearest the sun, the distances increase in an arithmetical ratio, 
w 7 hile those most remote increase in a geometrical ratio. Thus, 
if we add 30 to the distance of Mercury, it gives us nearly that of 
Venus ; 30 more gives that of the Earth ; while Saturn is nearly 
twice the distance of Jupiter, and Uranus twice the distance of 
Saturn. Between the orbits of Mars and Jupiter, a great chasm 
appeared, which broke the continuity of the series ; but the dis- 
covery of the new planets has filled the void. A more exact law 
of the series was discovered a few years since by Mr. Bode of 
Berlin. It is as follows : if we represent the distance of Mercury 
by 4, and increase each term by the product of 3 into a certain 
power of 2, we shall obtain the distances of each of the planets in 
succession. Thus, 



Mercury, . 


4 


- 4 


Venus, . 


4+3.2° 


= 7 


Earth, . 


4+3.2 1 


= 10 


Mars, 


4+3.2 2 


= 16 


Ceres, 


4+3.2 3 


= 28 


Jupiter, 


4+3.2 4 


= 52 


Saturn, 


4+3.2 5 


=100 


Uranus, 


4+3.2 6 


=196 



For example, by this law, the distances of the Earth and Jupi- 
ter are to each other as 10 to 52. Their actual distances as given 
in the table (Art. 299,) are as 1 to 5.202776 ; but 1 : 5.202776 : : 
10 : 52 nearly. 

The mean distances of the planets from the sun, may also be de- 
termined by means of Kepler's law, that the squares of the period- 



MAGNITUDES. 



177 



ical times are as the cubes of the distances, (Art. 192.) Thus the 
earth's distance being previously ascertained by means of the 
sun's horizontal parallax, (Art. 87,) and the period of any other 
planet as Jupiter, being learned from observation, we say as 
365T256 2 : 4332.585 2 * : : l 3 : 5.202 3 . But 5.202 is the number, 
which, according to the table, (Art. 299,) expresses the distance of 
Jupiter from the sun. 

300. MAGNITUDES. 



Diam. in Miles. Mean apparent Diam. Volume. 



3140 


6".9 


i 


7700 


16".9 


9 


7912 




1 


4200 


6".3 


l 

T 


160 


0".5 




89000 


36".7 


1281 


79000 


lf>".2 


995 


35000 


4".0 


80 



Mercury, 
Venus, . 
Earth, . 

Mars, 
Ceres, . 
Jupiter, . 
Saturn . 
Uranus . 



We remark here a great diversity in regard to magnitude, a 
diversity which does not appear to be subject to any definite 
law. While Venus, an inferior planet, is T 9 „- as large as the earth, 
Mars, a superior planet, is only 4, while Jupiter is 1281 times as 
large. Although several of the planets, when nearest to us, appear 
brilliant and large when compared with the fixed stars, yet the 
angle which they subtend is very small, that of Venus, the great- 
est of all, never exceeding about 1', or more exactly 61 ".2, and 
that of Jupiter being when greatest only about f of a minute. 

The distance of one of the near planets, as Venus or Mars, may 
be determined from its parallax ; and the distance being known, 
its real diameter can be estimated from its apparent diameter, in 
the same manner as we estimate the diameter of the sun. (Art. 
145.) 



* This is the number of days in one revolution of Jupiter. 
23 



178 





THE PLANETS. 






301. 


PERIODIC TIMES. 






Revolution 


in its orbit. 


Mean daily motion. 


Mercury 


3 months 


,or 


88 days, 


4° 5' 32".6 


Venus, 


7J " 


u 


224 « 


1° 36' 7".8 


Earth, 


1 year, 


it 


365 " 


0° 59' 8".3 


Mars, 


2 " 


u 


687 " 


0° 31' 26".7 


Ceres, 


4 " 


a 


1681 " 


0° 12' 50".9 


Jupiter, 


12 " 


u 


4332 " 


0° 4' 59".3 


Saturn, 


29 " 


u 


10759 " 


0° 2' 0".6 


Uranus, 


84 " 


a 


30686 " 


0° 0' 42".4 



From this view, it appears that the planets nearest the sun move 
most rapidly. Thus Mercury performs nearly 350 revolutions 
while Uranus performs one. This is evidently not owing merely 
to the greater dimensions of the orbit of Uranus, for the length of 
its orbit is not 50 times that of the orbit of Mercury, while the 
time employed in describing it is 350 times that of Mercury. In- 
deed this ought to follow from Kepler's law that the squares of 
the periodical times are as the cubes of the distances, from which 
it is manifest that the times of revolution increase faster than the 
dimensions of the orbit. Accordingly, the apparent progress of 
the most distant planets is exceedingly slow, the daily rate of Ura- 
nus being only 42".4 per day ; so that for weeks and months, and 
even years, this planet but slightly changes its place among the 
stars. 

THE INFERIOR PLANETS MERCURY AND VENUS. 

302. The inferior planets, Mercury and Venus, having their or- 
bits so far within that of the earth, appear to us as attendants upon 
the sun. Mercury never appears further from the sun than 29° 
(28° 48') and seldom so far ; and Venus never more than about 
47° (47° 12'). Both planets, therefore, appear either in the west 
soon after sunset, or in the east a little before sunrise. In high 
latitudes, where the twilight is prolonged, Mercury can seldom be 
seen with the naked eye, and then only at the periods of its great- 
est elongation.* The reason of this will readily appear from the 
following diagram. 

* Copernicus is said to have lamented on his death-bed that he had never been able 
to obtain a sight of Mercury, and Delambre saw it but twice. 



INFERIOR PLANETS MERCURY AND VENUS. 



179 



Let S (Fig. 60,) represent the sun, ADB the orbit of Mercury, 
and E the place of the Earth. Each of the planets is seen at its 
greatest elongation, when a line, EA or EB in the figure, is a tan- 
gent to its orbit. Then the sun being at S' in the heavens, the 
planet will be seen at A' and B', when at its greatest elongations, 
and will appear no further from the sun than the arc S'A' or S'B' 
respectively. 

Fig. 60. 




303. A planet is said to be in conjunction with the sun, when it 
is seen in the same part of the heavens with the sun, or when it 
has the same longitude. Mercury and Venus have each two con- 
junctions, the inferior and the superior. The inferior conjunction 
is its position when in conjunction on the same side of the sun 
with the earth, as at C in the figure : the superior conjunction is its 
position when on the side of the sun most distant from the earth, 
as at D. 



304. The period occupied by a planet between two successive 
conjunctions with the earth, is called its synodical revolution. 
Both the planet and the earth being in motion, the time of the 
synodical revolution exceeds that of the sidereal revolution of 
Mercury or Venus; for when the planet comes round to the place 
where it before overtook the earth, it does not find the earth at 
that point, but far in advance of it. Thus, let Mercury come into 



180 THE PLANETS. 

inferior conjunction with the earth at C, (Fig. 60.) In about 88 
days, the planet will come round to the same point again; but 
meanwhile the earth has moved forward through the arc EE', and 
will continue to move while the planet is moving more rapidly to 
overtake her, the case being analogous to that of the hour and 
second hand of a clock. 

Having the sidereal period of a planet, (which may always be 
accurately determined by observation,) we may ascertain its sy- 
nodical period as follows. Let T denote the sidereal period of 
the earth, and T' that of the planet, Since, in the time T the 

T' 

earth describes a complete revolution, T : T' : : 1 : -*- = the part 

of the circumference described by the earth in the time T\ But 
during the same time the planet describes a whole circumference. 

T' 

Therefore, 1 — -^ is what the planet gains on the earth in one 

revolution. In order to a new conjunction the planet must gain 
an entire circumference ; therefore, denoting the synodical period 
by S, the gain in one revolution will be to the time in which it 
is acquired, as a whole circumference is to the time in which that 
is gained, which is the synodical period. That is, 

nrv TT' 

T'::1:S=: 



From this formula we may find the synodical revolution of Mer- 
cury or Venus, by substituting the numbers denoted by the letters. 

Thus, 365,256X87969 =1 15.877, which is the synodical period 

277.287 

of Mercury. 

By a similar computation, the synodical revolution of Venus 
will be found to be about 584 days. 

305. The motion of an inferior planet is direct in passing through 
its superior conjunction, and retrograde in passing through its infe- 
rior conjunction. Thus Venus, while going from B through D to 
A, (Fig. 60,) moves in the order of the signs, or from west to east, 
and would appear to traverse the celestial vault B'S'A' from right 
to left ; but in passing from A through C to B, her course would 
be retrograde, returning on the same arc from left to right. If 



INFERIOR PLANETS MERCURY AND VENUS. 181 

the earth were at rest, therefore, (and the sun, of course, at rest,) 
the inferior planets would appear to oscillate backwards and for- 
wards across the sun. But, it must be recollected, that the earth 
is moving in the same direction with the planet, as respects the 
signs, but with a slower motion. This modifies the motions of the 
planet, accelerating it in the superior and retarding it in the infe- 
rior conjunctions. Thus in figure 60, Venus while moving through 
BDA would seem to move in the heavens from B' to A' were the 
earth at rest ; but meanwhile the earth changes its position from 
E to E', by which means the planet is not seen at A' but at A", 
being accelerated by the arc A' A" in consequence of the earth's 
motion. On the other hand, when the planet is passing through 
its inferior conjunction ACB, it appears to move backwards in the 
heavens from A' to B' if the earth is at rest, but from A' to B" 
if the earth has in the mean time moved from E to E', being re- 
tarded by the arc B'B". Although the motions of the earth have 
the effect to accelerate the planet in the superior conjunction, and 
to retard it in the inferior, yet, on account of the greater distance, 
the apparent motion of the planet is much slower in the superior 
than in the inferior conjunction. 

306. When passing from the superior to the inferior conjunction, 
or from the inferior to the suj)erior conjunction, through the greatest 
elongations, the inferior planets are stationary. 

If the earth were at rest, the stationary points would be at the 
greatest elongations as at A and B, for then the planet would be 
moving directly towards or from the earth, and would be seen for 
some time in the same place in the heavens ; but the earth itself 
is moving nearly at right angles to the line of the planet's motion, 
that is, the line which is drawn from the earth to the planet through 
the point of greatest elongation ; hence a direct motion is given 
to the planet by this cause. When the planet, however, has passed 
this line, by its superior velocity it soon overcomes this tendency 
of the earth to give it a relative motion eastward, and becomes 
retrograde as it approaches the inferior conjunction. Its stationary 
point obviously lies between its place of greatest elongation, and 
the place where its motion becomes retrograde. Mercury is sta- 



182 THE PLANETS. 

tionary at an elongation of from 15° to 20° from the sun ; and 
Venus at about 29°.* 

307. Mercury and Venus exhibit to the telescope phases similar to 
those of the moon. 

When on the side of their inferior conjunctions, these planets 
appear horned, like the moon in her first and last quarters ; and 
when on the side of their superior conjunctions, they appear gib- 
bous. At the moment of superior conjunction, the whole enlight- 
ened orb of the planet is turned towards the earth, and the appear- 
ance would be that of the full moon, but the planet is too near the 
sun to be commonly visible. 

These different phases show that these bodies are opake, and 
shine only as they reflect to us the light of the sun ; and the same 
remark applies to all the planets. 

308. The distance of an inferior planet from the sun, may be 
found by observations at the time of its greatest elongation. 

Thus if E be the place of the earth, and B that of Venus at the 
time of her greatest elongation, the angle SBE will be known, 
being a right angle. Also the angle SEB is known from observa- 
tion. Hence the ratio of SB to SE becomes known ; or, since SE 
is given, being the distance of the earth from the sun, SB the radius 
of the orbit of the planet is determined. If the orbits were both 
circles, this method would be very exact ; but being elliptical, we 
obtain the mean value of the radius SB by observing its greatest 
elongation in different parts of its orbit, f 

309. The orbit of Mercury is the most eccentric, and the most 
inclined of all the planets ; J while that of Venus varies but little 
from a circle, and lies much nearer to the ecliptic. 

The eccentricity of the orbit of Mercury is nearly £ its semi- 
major axis, while that of Venus is only T |j ; the inclination of 
Mercury's orbit is 7°, while that of Venus is less than 3°£.§ Mer- 
cury, on account of his different distances from the earth, varies 

* Herschel, p. 242.— Woodhouse, 557. t Herschel, p. 239. 

* The new planets are of course excepted. § Baily's Tables. 



INFERIOR PLANETS MERCURY AND VENUS. 183 

much in his apparent diameter, which is only 5" in the apogee, 
but 12" in the perigee, 

310. The most favorable time for determining the sidereal revo- 
lution of a planet, is when its conjunction takes place at one of 
its nodes ; for then the sun, the earth, and the planet, being in the 
same straight line, it is referred to its true place in the heavens, 
whereas, in other positions, its apparent place is more or less 
affected by perspective. 

311. An inferior planet is brightest at a certain point between 
its greatest elongation and inferior conjunction. 

Its maximum brilliancy would happen at the inferior conjunc- 
tion, (being then nearest to us,) if it shined by its own light ; 
but in that position, its dark side is turned towards us. Still, its 
maximum cannot be when most of the illuminated side is towards 
us ; for then, being at the superior conjunction, it is at its greatest 
distance from us. The maximum must therefore be somewhere 
between the two. Venus gives her greatest light when about 40° 
from the sun. 

312. Mercury and Venus both revolve on their axes, in nearly the 
same time with the earth. 

The diurnal period of Mercury is 24h. 5m. 28s., and that of 
Venus 23h. 21m. 7s. The revolutions on their axes have been 
determined by means of some spot or mark seen by the telescope, 
as the revolution of the sun on his axis is ascertained by means of 
his spots. 

313. Venus is regarded as the most beautiful of the planets, and 
is well known as the morning and evening star. The most ancient 
nations did not indeed recognize the evening and morning star as 
one and the same body, but supposed they were different planets, 
and accordingly gave them different names, calling the morning 
star Lucifer, and the evening star Hesperus. At her period of 
greatest splendor, Venus casts a shadow, and is sometimes visible 
in broad daylight. Her light is then estimated as equal to that of 



184 



THE PLANETS. 



twenty stars of the first magnitude.* At her period of greatest 
elongation, Venus is visible from three to four hours after the set- 
ting or before rising of the sun. 

314. Every eight years, Venus forms her conjunctions with the 
sun in the same part of the heavens. 

For, since the synod ical period of Venus is 584 days, and her 
sidereal period 224.7, 

224.7 : 360°:: 584 : 935.6=the arc of longitude described by 
Venus between the first and second conjunctions. Deducting 720°, 
or two entire circumferences, the remainder, 215°.6, shows how 
far the place of the second conjunction is in advance of the first. 
Hence, in five synodical revolutions, or 2920 days, the same point 
must have advanced 215°.6x5=1078°, which is nearly three 
entire circumferences, so that at the end of five synodical revolu- 
tions, occupying 2920 days, or 8 years, the conjunction of Venus 
takes place nearly in the same place in the heavens as at first. 

Whatever appearances of this planet, therefore, arise from its 
positions with respect to the earth and the sun, they are repeated 
every eight years in nearly the same form. 

TRANSITS OF THE INFERIOR PLANETS. 

315. The Transit of Mercury or Venus, is its passage across the 
sun's disk, as the moon passes over it in a solar eclipse. 

As a transit takes place only when the planet is in inferior con- 
junction, at which time her motion is retrograde (Art. 305,) it is 
always from left to right, and the planet is seen projected on the 
solar disk in a black round spot. Were the orbits of the inferior 
planets coincident with the plane of the earth's orbit, a transit 
would occur to some part of the earth at every inferior conjunc- 
tion. But the orbit of Venus makes an angle of 3£° with the 
ecliptic, and Mercury an angle of 7° ; and, moreover, the apparent 
diameter of each of these bodies is very small, both of which cir- 
cumstances conspire to render a transit a comparatively rare 
occurrence, since it can happen only when the sun, at the time of 



* Francoeur, Uranography, p. 125. 



TRANSITS OP THE INFERIOR PLANETS. 185 

an inferior conjunction, chances to be at or extremely near the 
planet's node. The nodes of Mercury lie in longitude 46° and 
226°, points which the sun passes through in May and November. 
It is only in these months, therefore, that transits of Mercury can 
occur. For a similar reason, those of Venus occur only in June 
and December. Since, however, the nodes of both planets have 
a small retrograde motion, the months in which transits occur will 
change in the course of ages. 

316. The intervals between successive transits, may be found in 

the following manner. The formula which gives the synodical 

TxT' 
period (Art. 304,) is S= , where S denotes the period, T the 

sidereal revolution of the earth, and T ; that of the planet. If we 
now represent by m the number of synodical periods of the sun* 
in the required interval, and by n the number of synodical periods 
of the planet ; then, since the number of periods in each case is in- 

TxT' m T' 

versely as the time of one, we have, T : - — — : : n:m /.—. 



T-T' n~T-T' 

In the case of Mercury, whose sidereal period is 87.969 days, w r hile 

that of the earth is, 365.256 days, — — — ; that is, after the 

n 277287 

earth has revolved 87969 times, (or after this number of years,) 
Mercury will have revolved just 277287, and the two bodies will 
be together again at the place where they started. But as periods 
of such enormous length do not fall within the observation of 
man, let us search for smaller numbers having nearly the same 
ratio. Now, 

87969 : 365256 : : 1 : 4} (nearly.) 
This shows that in one year Mercury will have made 4 revo- 
lutions and i of another ; so that, when the sun returns to the same 
node, Mercury will be more than 60° in advance of it ; conse- 
quently, no transit can take place after an interval of one year. 
But, by making trial of 2, 3, 4, &c. years, we shall find a nearer 
approximation at the end of 6 years ; for, 

\ 

* That is, the time in which the sun returns again to the planet's node, which is ob- 
viously after one year. 

24 



186 THE PLANETS. 

87969 : 365265 :; 6 : 25- T V- In 6 years, therefore, Mercury 
will fall short of reaching the node by only T V of a revolution, or 
about 33°. In ]3 years the chance of meeting will be much 
greater, for in this period the earth will have made 1 3 and Mer- 
cury 54 revolutions. The numbers 33 and 137, 46 and 191, afford 
a still nearer approximation.* 

317. In a similar manner, transits of Venus are probable after 
8, 227, 235, and 243 years. Since Venus returns to her conjunc- 
tion at nearly the same point of her orbit, after 8 years, (Art. 314,) 
it frequently happens that a transit takes place after an interval 
of 8 years. But at that time Venus is so far from her node, that 
her latitude amounts to from 20' to 24'. Still she may possibly 
come within the sun's disk as she passes by him ; for suppose at 
the preceding transit her latitude was 10' on one side of the node 
and is now 10' on the other side, this being less than the sun's 
semi-diameter, a transit may occur 8 years after another. Thus 
transits of Venus took place in 1761 and 1769. But in 16 years 
the latitude changes from 40' to 48', and Venus could not reach 
any part of the solar disk in her inferior conjunction. 

From the above series we should infer that another transit 
could not take place under 227 years ; but since there are two 
nodes, the chance is doubled, so that a transit may occur at the 
other node in half that interval, or in about 113 years. If, at the 
occurrence of the first transit, Venus had passed her node, the 
next transit at the other node will happen 8 years before the 113 
are completed ; or if she had not reached the node, it will happen 
8 years later. Hence, after two transits have occurred within 8 
years, another cannot be expected before 105, 113, or 121 years. 
Thus, the next transit will happen in 1874=1769+105; also in 
1882=1874+8. 

318. The great interest attached by astronomers to a transit of 
Venus, arises from its furnishing the most accurate means in our 
power of determining the sun's horizontal parallax, — an element 
of great importance, since it leads us to a knowledge of the distance 

* This series may readily be obtained by the method of Continued Fractions. See 
Davies's Bourdon's Algebra. 



TRANSITS OF THE INFERIOR PLANETS. 187 

of the earth from the sun, and consequently, by the application 
of Kepler's law, (Art. 183,) of the distances of all the other planets. 
Hence, in 1769, great efforts were made throughout the civilized 
world, under the patronage of different governments, to observe 
this phenomenon under circumstances the most favorable for de- 
termining the parallax of the sun. 

The method of finding the parallax of a heavenly body described 
in article 85, cannot be relied on to a greater degree of accuracy 
than 4". In the case of the moon, whose greatest parallax amounts 
to about 1°, this deviation from absolute accuracy is not material ; 
but it amounts to nearly half the entire parallax of the sun. 

319. If the sun and Venus were equally distant from us, they 
would be equally affected by parallax as viewed by spectators in 
different parts of the earth, and hence their relative situation would 
not be altered by it; but since Venus, at the inferior conjunction, 
is only about one third as far off as the sun, her parallax is propor- 
tionally greater, and therefore spectators at distant points will see 
Venus projected on different parts of the solar disk, and as the 
planet traverses the disk, she will appear to describe chords of dif- 
ferent lengths, by means of which the duration of the transit may 
be estimated at different places. The difference in the duration 
of the transit does not amount to many minutes ; but to make it 
as large as possible very distant places are selected for observation. 
Thus in the transit of 1769, among the places selected, two of the 
most favorable were Wardhuz in Lapland, and Oteheite,* one of 
the South Sea Islands. 

The principle on which the sun's horizontal parallax is estimated 
from the transit of Venus, may be illustrated as follows: Let E 
(Fig. 61,) be the earth, V Venus, and S the sun. Suppose A, B, 
two spectators at opposite extremities of that diameter of the earth 
which is perpendicular to the ecliptic. The spectator at A will 
see Venus on the sun's disk at a, and the spectator at B will see 
Venus at b ; and since AV and BV may be considered as equal 
to each other, as also Vb and Va, therefore the triangles AVB and 
Vab are similar to each other, and AV : Va :: AB : ab. But the 
ratio of AV to Va is known, (Art. 308) ; hence, the ratio of AB to 

* Now written Tahiti. 



188 THE PLANETS. 

ab is known, and when the angular value of ah as seen from the 
earth, is found, that of AB becomes known, as seen from the sun ;* 
and half AB, or the semi-diameter of the earth as seen from the 



Fig. 61. 




sun, is the sun's horizontal parallax. To find the apparent diameter 
of ab, we have only to find the breadth of the space between the 
two chords. Now, we can ascertain the value of each chord by 
the time occupied in describing it, since the motions of Venus and 
those of the sun are accurately known from the tables. Each 
chord being double the sine of half the arc cut off by it, therefore 
the sine of half the arc and of course the versed sine becomes 
known, and the difference of the two versed sines is the breadth 
of the zone in question. There are many circumstances to be 
taken into the account in estimating, from observations of this 
kind, the sun's horizontal parallax ; but the foregoing explanation 
may be sufficient to give the learner an idea of the general princi- 
ples of this method. The appearance of Venus on the sun's disk, 
being that of a well defined black spot, and the exactness with 
which the moment of external or internal contact may be deter- 
mined, are circumstances favorable to the exactness of the result ; 
and astronomers repose so much confidence in the estimation of 
the sun's horizontal parallax as derived from the observations on 
the transit of 1769, that this important element is thought to be 
ascertained within T V of a second. The general result of all these 
observations give the sun's horizontal parallax 8."6, or more ex- 
actly, 8."5776.f 

* If, for example, ab is 2^ times AB, (which is nearly the fact,) then if AB were on 

the sun instead of on the earth, it would subtend an angle at the eye equal to — of ab. 

But if viewed from the sun, the distance being the same, its apparent diameter must be 
the same. 

tDelambre,t.2. Vince's Complete Syst. vol. 1. Woodhouse, p. 754. Herschel,p.243. 



SUPERIOR PLANETS. 189 

320. During the transits of Venus over the sun's disk in 1761 
and 1769, a sort of penumbral light was observed around the 
planet by several astronomers, which was thought to indicate an 
atmosphere. This appearance was particularly observable while 
the planet was coming on and going off the solar disk. The total 
immersion and emersion were not instantaneous ; but as two drops 
of water, when about to separate, form a ligament between them, 
so there was a dark shade stretched out between Venus and the 
sun, and when the ligament broke, the planet seemed to have got 
about an eight part of her diameter from the limb of the sun.* 
The various accounts of the two transits abound with remarks like 
these, which indicate the existence of an atmosphere about Venus 
of nearly the density and extent of the earth's atmosphere. Similar 
proofs of the existence of an atmosphere around this planet, are 
derived from appearances of twilight. 



CHAPTER X 



OF THE SUPERIOR PLANETS MARS, JUPITER, SATURN, AND URANUS. 

■321. The Superior planets are distinguished from the Inferior, 
by being seen at all distances from the sun from 0° to 180°. 
Having their orbits exterior to that of the earth, they of course 
never come between us and the sun, that is, they never have any 
inferior conjunction like Mercury and Venus, but they are some- 
times seen in superior conjunction, and sometimes in opposition. 
Nor do they, like the inferior planets, exhibit to the telescope dif- 
ferent phases, but, with a single exception, they always present 
the side that is turned towards the earth fully enlightened. This 
is owing to their great distance from the earth ; for were the spec- 
tator to stand upon the sun, he would of course always have the 
illuminated side of each of the planets turned towards him ; but, 
so distant are all the superior planets except Mars, that they are 

* Edinb, Encyc. Art. Astronomy. 



190 THE PLANETS. 

viewed by us very nearly in the same manner as they would be if 
we actually stood on the sun. 

322. Mars is a small planet, his diameter being only about half 
that of the earth, or 4100 miles. He also, at times, comes nearer to 
the earth than any other planet except Venus. His mean distance 
from the sun is 142,000,000 miles; but his orbit is so eccentric 
that his distance varies much in different parts of his revolution. 
Mars is always very near the ecliptic, never varying from it 2°. 
He is distinguished from all the other planets by his deep red color, 
and fiery aspect ; but his brightness and apparent magnitude vary- 
much at different times, being sometimes nearer to us than at 
others, by the whole diameter of the earth's orbit, that is, by about 
190,000,000 of miles. When Mars is on the same side of the sun 
with the earth, or at his opposition, he comes within 47,000,000 
miles of the earth, and rising about the time the sun sets surprises 
us by his magnitude and splendor; but when he passes to the 
other side of the sun to his superior conjunction, he dwindles to 
the appearance of a small star, being then 237,000,000 miles from 
us. Thus, let M (Fig. 62,) represent Mars in opposition, and M' 




M 



in the superior conjunction. It is obvious that in the former situa- 
tion, the planet must be nearer to the earth than in the latter by 
the whole diameter of the earth's orbit. 



JUPITER. 191 

323. Mars is the only one of the superior planets which exhibits 
phases. When he is towards the quadratures at Q or Q', it is 
evident from the figure that only a part of the circle of illumina- 
tion is turned towards the earth, such a portion of the remoter 
part of it being concealed from our view as to render the form 
more or less gibbous. 

324. When viewed with a powerful telescope, the surface of 
Mars appears diversified with numerous varieties of light and 
shade. The region around the poles is marked by white spots, 
which vary their appearance with the changes of seasons in the 
planet. Hence Dr. Herschel conjectured that they were owing 
to ice and snow, which alternately accumulates and melts, accord- 
ing to the position of each pole with respect to the sun.* It has 
been common to ascribe the ruddy light of this planet to an exten- 
sive and dense atmosphere, which was said to be distinctly indi- 
cated, by the gradual diminution of light observed in a star as it 
approached very near to the planet in undergoing an occultation ; 
but more recent observations afford no such evidence of an atmo- 
sphere, f 

By observations on the spots we learn that Mars revolves on his 
axis in very nearly the same time with the earth, (24h. 39m. 2P.3); 
and that the inclination of his axis to that of the ecliptic is also 
nearly the same, being 30° 18' 10".8.J 

325. As the diurnal rotation of Mars is nearly the same as that 
of the earth, we might expect a similar flattening at the poles, 
giving to the planet a spheroidal figure. Indeed the compression 
or ellipticity of Mars greatly exceeds that of the earth, being no 
less than T \ of the equatorial diameter, while that of the earth is 
on ty 30T5 (Art. 138.) This remarkable flattening of the poles of 
Mars has been supposed to arise from a great variation of density 
in the planet in different parts.§ 

326. Jupiter is distinguished from all the other planets by his 
vast magnitude. His diameter is 89,000 miles, and his volume 

* Phil. Trans. 1784. t Sir James South, Phil. Trans. 1833. 

% Baily's Tables, p. 29. § Ed. Encyc. Art. Astronomy. 



192 THE PLANETS. 

1280 times that of the earth. His figure is strikingly spheroidal, 
the equatorial being larger than the polar diameter in the propor- 
tion of 107 to 100. (See Frontispiece, Fig. 4.) Such a figure 
might naturally be expected from the rapidity of his diurnal rota- 
tion, which is accomplished in about 10 hours. A place on the 
equator of Jupiter must turn 27 times as fast as on the terrestrial 
equator. The distance of Jupiter from the sun is nearly 490,000,000 
miles, and his revolution around the sun occupies nearly 12 
years. 

327. The view of Jupiter through a good telescope, is one of 
the most magnificent and interesting spectacles in astronomy. 
The disk expands into a large and bright orb like the full moon ; 
the spheroidal figure which theory assigns to revolving spheres, is 
here palpably exhibited to the eye ; across the disk, arranged in 
parallel stripes, are discerned several dusky bands, called belts ; 
and four bright satellites, always in attendance, but ever varying 
their positions, compose a splendid retinue. Indeed, astronomers 
gaze with peculiar interest on Jupiter and his moons as affording 
a miniature representation of the whole solar system, repeating on 
a smaller scale, the same revolutions, and exemplifying, in a man- 
ner more within the compass of our observation, the same laws as 
regulate the entire assemblage of sun and planets. (See Fig. 63.) 

328. The Belts of Jupiter, are variable in their number and di- 
mensions. With the smaller telescopes, only one or two are seen 
across the equatorial regions ; but with more powerful instruments, 
the number is increased, covering a great part of the whole disk. 
Different opinions have been entertained by astronomers respect- 
ing the cause of the belts ; but they have generally been regarded 
as clouds formed in the atmosphere of the planet, agitated by 
winds, as is indicated by their frequent changes, and made to as- 
sume the form of belts parallel to the equator by currents that cir- 
culate around the planet like the trade winds and other currents 
that circulate around our globe.* Sir John Herschel supposes 
that the belts are not ranges of clouds, but portions of the planet 
itself brought into view by the removal of clouds and mists 

* Ed. Encyc. Art. Astronomy. 



JUPITER. 



193 



that exist in the atmosphere of the planet through which are open- 
ings made by currents circulating around Jupiter.* 

329. The Satellites of Jupiter may be seen with a telescope of 
very moderate powers. Even a common spy-glass will enable us 
to discern them. Indeed one or two of them have been occasion- 
ally seen with the naked eye. In the largest telescopes, they sev- 
erally appear as bright as Sirius. With such an instrument the 
view of Jupiter with his moons and belts is truly a magnificent 
spectacle, a world within itself. As the orbits of the satellites do 
not deviate far from the plane of the ecliptic, and but little from 
the equator of the planet, they are usually seen in nearly a straight 
line with each other extending across the central part of the disk. 
(See Frontispiece.) 

330. Jupiter's satellites are distinguished from one another by 
the denominations of first, second, third, and fourth, according to 
their relative distances from Jupiter, the first being that which is 
nearest to him. Their apparent motion is oscillatory, like that of 
a pendulum, going alternately from their greatest elongation on 
one side to their greatest elongation on the other, sometimes in a 
straight line, and sometimes in an elliptical curve, according to the 
different points of view in which we observe them from the earth. 
They are sometimes stationary ; their motion is alternately direct 
and retrograde ; and, in short, they exhibit in miniature all the 
phenomena of the planetary system. Various particulars of the 
system are exhibited in the following table. The distances are 
given in radii of the primary. 



Satellite. 


Diameter. 


Mean Distance. 


Sidereal Revolution. 


1 
2 
3 
4 


2508 
2068 
3377 
2890 


6.04853 

9.62347 

15.35024 

26.99835 


Id. 18h. 28m. 
3 13 14 
7 3 43 
16 16 32 



Hence it appears, first, that Jupiter's satellites are all somewhat 
larger than the moon, (except the second, which is very nearly 
the size of the moon,) and the third the largest of the whole, but the 

* Herschel's Astron. p. 266. 
25 



194 THE PLANETS. 

diameter of the latter is only about ^\ part of that of the primary ; 
secondly, that the distance of the innermost satellite from the 
planet is three times his diameter, while that of the outermost 
satellite is nearly fourteen times his diameter ; thirdly, that the 
first satellite completes its revolution around the primary in one 
day and three fourths, while the fourth satellite requires nearly 
sixteen and three fourths days. 

331. The orbits of the satellites are nearly or quite circular, and 
deviate but little from the plane of the planet's equator, and of 
course are but slightly inclined to the plane of his orbit. They 
are, therefore, in a similar situation with respect to Jupiter as the 
moon would be with respect to the earth if her orbit nearly coin- 
cided with the ecliptic, in which case she would undergo an eclipse 
at every opposition. 

332. The eclipses of Jupiter's satellites, in their general concep- 
tion, are perfectly analogous to those of the moon, but in their de- 
tail they differ in several particulars. Owing to the much greater 
distance of Jupiter from the sun, and its greater magnitude, the 
cone of its shadow is much longer and larger than that of the 
earth, (Art. 248.) On this account^ as well as on account of the 
little inclination of their orbits to that of their primary, the three 
inner satellites of Jupiter pass through the shadow, and are totally 
eclipsed at every revolution. The fourth satellite, owing to the 
greater inclination of its orbit, sometimes though rarely escapes 
eclipse, and sometimes merely grazes the limits of the shadow or 
suffers a partial eclipse.* These eclipses, moreover, are not seen 
as is the case with those of the moon, from the center of their mo- 
tion, but from a remote station, and one whose situation with re- 
spect to the line of the shadow is variable. This, of course, makes 
no difference in the times of the eclipses, but a very great one in 
their visibility, and in their apparent situations with respect to the 
planet at the moment of their entering or quitting the shadow. 

333. The eclipses of Jupiter's satellites present some curious 
phenomena, which will be understood from the following diagram. 

* Sir J. Herschel, Ast. p. 276. 



JUPITER. 195 

Let A, B, C, D, (Fig. 63,) represent the earth in different parts of 
its orbit ; J, Jupiter in his orbit MN, surrounded by his four satel- 
lites, the orbits of which are marked 1, 2, 3, 4. At a the first 
satellite enters the shadow of the planet, and emerges from it at 
b, and advances to its greatest elongation at c. Since the shadow 
is always opposite to the sun, only the immersion of a satellite 
will be visible to the earth while the earth is somewhere between 

Fig. 63. 




C and A, that is, while the earth is passing from the position 
where it has the planet in superior conjunction, to that where it 
has the planet in opposition ; for while the earth is in this situation, 
the planet conceals from its view the emersion, as is evident from 
the direction of the visual ray/c/. For a similar reason the emer- 
sion only is visible while the earth passes from A to C, or from 
the opposition to the superior conjunction. In other words, when 
the earth is to the westward of Jupiter, only the immersions of a 
satellite are visible ; when the earth is to the eastward of Jupiter, 
only the emersions are visible. This, however, is strictly true only 
of the first satellite ; for the third and fourth, and sometimes even 
the second, owing to their greater distances from Jupiter, occa- 
sionally disappear and reappear on the same side of the disk. 
The reason why they should reappear on the same side of the 
disk, will be understood from the figure. Conceive the whole sys- 
tem of Jupiter and his satellites as projected on the more distant 
concave sphere, by lines drawn, like/d, from the observer on the 
earth at D through the planet and each of the satellites ; then it is 
evident that the remoter parts of the shadow where the exterior sat- 
ellites traverse it, will fall to the westward of the planet, and of 



196 THE PLANETS. 

course these satellites as they emerge from the shadow will be pro- 
jected to a point on the same side of the disk as the point of their 
immersion. The same mode of reasoning will show that when 
the earth is to the eastward of the planet, the immersions and 
emersions of the outermost satellites will be both seen on the east 
side of the disk. When the earth is in either of the positions C 
or A, that is, at the superior conjunction or opposition of the planet, 
both the immersions and emersions take place behind the planet, 
and the eclipses occur close to the disk. 

334. When one of the satellites is passing between Jupiter and 
the sun, it casts a shadow upon its primary, which is seen by the 
telescope travelling across the disk of Jupiter, as the shadow of the 
moon would be seen to traverse the earth by a spectator favor- 
ably situated in space. When the earth is to the westward of Ju- 
piter, as at D, the shadow reaches the disk of the planet, or is seen 
on the disk, before the satellite itself reaches it. For the satellite 
will not enter on the disk, until it comes up to the line fd at d, a 
point which it reaches later than its shadow reaches the same line. 
After the earth has passed the opposition, as at B, then the satel- 
lite will reach the visual ray at d sooner than the shadow, and 
of course be sooner projected on the disk. In the transits of Ju- 
piter's satellites, which with very powerful telescopes may be ob- 
served with great precision, the satellite itself is sometimes seen on 
the disk as a bright spot, if it chances to be projected upon one of 
the belts. Occasionally, also, it is seen as a dark spot, of smaller 
dimensions than the shadow. This curious fact has led to the 
conclusion, that certain of the satellites have sometimes on their 
own bodies or in their atmospheres, obscure spots of great extent.*" 

335. A very singular relation subsists between the mean motions 
of the three first satellites of Jupiter. The mean longitude of the 
first satellite, minus three times that of the second, plus twice that of 
the third, always equals 180 degrees. A curious consequence of 
this relation is, that the three satellites can never be all eclipsed at 
the same time ; for then their longitudes would be equal, and of 

* Sir J. Herschel. 



JUPITER. 1 97 

course the sum of their longitudes would be nothing.* It will be 
remarked, that these phenomena are such as would present them- 
selves to a spectator on Jupiter, and not to a spectator on the 
earth. 

336. The eclipses of Jupiter's satellites have been studied with 
great attention by astronomers, on account of their affording one 
of the easiest methods of determining the longitude. On this 
subject Sir J. Herschel remarks :f The discovery of Jupiter's 
satellites by Galileo, which was one of the first fruits of the inven- 
tion of the telescope, forms one of the most memorable epochs in 
the history of astronomy. The first astronomical solution of the 
great problem of " the longitude," — the most important problem 
for the interests of mankind that has ever been brought under the 
dominion of strict scientific principles, dates immediately from 
their discovery. The final and conclusive establishment of the 
Copernican system of astronomy, may also be considered as refer- 
able to the discovery and study of this exquisite miniature system, 
in which the laws of the planetary motions, as ascertained by 
Kepler, and especially that which connects their periods and dis- 
tances, were speedily traced, and found to be satisfactorily main- 
tained. 

337. The entrance of one of Jupiter's satellites into the shadow 
of the primary being seen like the entrance of the moon into the 
earth's shadow, at the same moment of absolute time, at all 
places where the planet is visible, and being wholly independent of 
parallax ; being, moreover, predicted beforehand with great accu- 
racy for the instant of its occurrence at Greenwich, and given in 
the Nautical Almanac ; this would seem to be one of those events 
(Art. 273,) which are peculiarly adapted for finding the longitude. 
It must be remarked, however, that the extinction of light in the 
satellite at its immersion, and the recovery of its light at its emer- 
sion, are not instantaneous, but gradual ; for the satellite, like the 
moon, occupies some time in entering into the shadow or in 
emerging from it, which occasions a progressive diminution or in- 

* Biot, Ast. Phys. t Elements of Ast. p. 279. 



198 THE PLANETS. 

crease of light. The better the light afforded by the telescope 
with which the observation is made, the later the satellite will be 
seen at its immersion, and the sooner at its emersion.* In noting 
the eclipses even of the first satellite, the time must be considered 
as uncertain to the amount of 20 or 30 seconds ; and those of the 
other satellites involve still greater uncertainty. Two observers, 
in the same room, observing with different telescopes the same 
eclipse, will frequently disagree in noting its time to the amount 
of 15 or 20 seconds ; and the difference will be always the same 
way.f 

Better methods, therefore, of finding the longitude are now 
employed, although the facility with which the necessary observa- 
tions can be made, and the little calculation required, still render 
this method eligible in many cases where extreme accuracy is not 
important. As a telescope is essential for observing an eclipse of 
one of the satellites, it is obvious that this method cannot be prac- 
ticed at sea. 

338. The grand discovery of the progressive motion of light, 
was first made by observations on the eclipses of Jupiter's satel- 
lites. In the year 1675, it was remarked by Roemer, a Danish 
astronomer, on comparing together observations of these eclipses 
during many successive years, that they take place sooner by 
about sixteen minutes (16m. 26 s . 6) when the earth is on the 
same side of the sun with the planet, than when she is on the op- 
posite side. This difference he ascribed to the progressive motion 
of light, which takes that time to pass through the diameter of the 
earth's orbit, making the velocity of light about 192,000 miles per 
second. So great a velocity startled astronomers at first, and pro- 
duced some degree of distrust of this explanation of the phenome- 
non ; but the subsequent discovery of the aberration of light (Art. 
195,) led to an independent estimation of the velocity of light 
with almost precisely the same result. 

339. Saturn comes next in the series as we recede from the 

* This is the reason why observers are directed in the Nautical Almanac to use tele- 
scopes of a certain power. 
t Woodhouse, p. 840. 



SATURN. 199 

sun, and has, like Jupiter, a system within itself, on a scale of great 
magnificence. In size it is, next to Jupiter, the largest of the 
planets, being 79,000 miles in diameter, or about 1,000 times as 
large as the earth. It has likewise belts on its surface and is at- 
tended by seven satellites. But a still more wonderful appendage 
is its Ring, a broad wheel encompassing the planet at a great dis- 
tance from it. We have already intimated that Saturn's system 
is on a grand scale. As, however, Saturn is distant from us nearly 
900,000,000 miles, we are unable to obtain the same clear and 
striking views of his phenomena as we do of the phenomena of 
Jupiter, although they really present a more wonderful mechanism. 

340. Saturn's ring, when viewed with telescopes of a high 
power, is found to consist of two concentric rings,* separated from 
each other by a dark space. (See Frontispiece.) Although this 
division of the rings appears to us, on account of our immense dis- 
tance, as only a fine line, yet it is in reality an interval of not less 
than about 1800 miles. The dimensions of the whole system are 
in round numbers, as follows :f 

Miles. 

Diameter of the planet, .... 79,000 

From the surface of the planet to the inner ring, 20,000 
Breadth of the inner ring, .... 17,000 
Interval between the rings, . . . 1,800 

Breadth of the outer ring, .... 10,500 
Extreme dimensions from outside to outside, 176,000 

The figure represents Saturn as it appears to a powerful tele- 
scope, surrounded by its rings, and having its body striped with 
dark belts, somewhat similar but broader and less strongly marked 
than those of Jupiter, and owing doubtless to a similar cause. 
That the ring is a solid opake substance, is shown by its throwing 
its shadow on the body of the planet on the side nearest the sun 
and on the other side receiving that of the body. From the par- 
allelism of the belts with the plane of the ring, it may be conjec- 
tured that the axis of rotation of the planet is perpendicular to 

* It is said that several additional divisions of the ring have been detected. — (Kater, 
Ast. Trans, iv. 383.) t Prof. Struve, Mem. Ast. Soc, 3. 301. 



200 THE PLANETS. 

that plane ; and this conjecture is confirmed by the occasional 
appearance of extensive dusky spots on its surface, which when 
watched indicate a rotation parallel to the ring in lOh. 29m. 17s. 
This motion, it will be remarked, is nearly the same with the diur- 
nal motion of Jupiter, subjecting places on the equator of the 
planet to a very swift revolution, and occasioning a high degree 
of compression at the poles, the equatorial being to the polar di- 
ameter in the high ratio of 11 to 10. But it is remarkable that the 
globe of Saturn appears to be flattened at the equator as well as 
at the poles. The polar compression extends to a great distance 
over the surface of the planet, and the greatest diameter is that of 
the parallel of 43° of latitude. The disk of Saturn, therefore, re- 
sembles a square of which the four corners have been rounded off.* 
It requires a telescope of high magnifying powers and a strong 
light to give a full and striking view of Saturn with his rings and 
belts and satellites ; for we must bear in mind that at the distance 
of Saturn one second of angular measurement corresponds to 4,000 
miles, a space equal to the semi-diameter of our globe. But with 
a telescope of moderate powers, the leading phenomena of the 
ring itself may be observed. 

341. Saturn's ring, in its revolution around the sun, always re- 
mains parallel to itself. 

If we hold opposite to the eye a circular ring or disk like a 
piece of coin, it will appear as a complete circle when it is at right 
angles to the axis of vision, but when oblique to that axis it will 
be projected into an ellipse more and more flattened as its obliquity 
is increased, until, when its plane coincides with the axis of vision, 
it is projected into a straight line. Let us place on the table a 
lamp to represent the sun, and holding the ring at a certain dis- 
tance inclined a little towards the lamp, let us carry it round the 
lamp, always keeping it parallel to itself. During its revolution it 
will twice present its edge to the lamp at opposite points, and 
twice at places 90° distant from those points, it will present its 
broadest face towards the lamp. At intermediate points, it will 
exhibit an ellipse more or less open, according as it is nearer one 

* Sir W. Herschel, Phil. Tr. 1806, Part II. 



SATURN. 



201 



or the other of the preceding positions. It will be seen also that 
in one half of the revolution the lamp shines on one side of the 
ring, and in the other half of the revolution on the other side. 
Such would be the successive appearances of Saturn's ring to a 
spectator on the sun ; and since the earth is, in respect to so dis- 
tant a body as Saturn, very near the sun, those appearances are 
presented to us in nearly the same manner as though we viewed 
them from the sun. Accordingly, we sometimes see Saturn's ring 
under the form of a broad ellipse, which grows continually more 
and more acute until it passes into a line, and we either lose sight 
of it altogether, or with the aid of the most powerful telescopes, 
we see it as a fine thread of light drawn across the disk and pro- 
jecting out from it on each side. As the whole revolution occupies 
30 years, and the edge is presented to the sun twice in the revolu- 
tion, this last phenomenon, namely, the disappearance of the ring, 
takes place every 15 years. 

342. The learner may perhaps gain a clearer idea of the fore- 
going appearances from the following diagram : 

Let A, B, C, &c. represent successive positions of Saturn and 
his ring in different parts of his orbit, while ab represents the 
orbit of the earth.* Were the ring when at C and G perpendicu- 

Fig. 64. 




lar to the line CG, it would be seen by a spectator situated at a 
or b a perfect circle, but being inclined to the line of vision 28° 4', 



* It may be remarked by the learner, that these orbits are made so elliptical, not to 
represent the eccentricity of either the earth's or Saturn's orbit, but merely as the pro- 
jection of circles seen very obliquely. 

26 



202 THE PLANETS. 

it is projected into an ellipse. This ellipse contracts in breadth 
as the ring passes towards its nodes at A and E, where it dwindles 
into a straight line. From E to G the ring opens again, becomes 
broadest at G, and again contracts till it becomes a straight line at 
A, and from this point expands till it recovers its original breadth 
at C. These successive appearances are all exhibited to a telescope 
of moderate powers. The ring is extremely thin, since the small- 
est satellite, when projected on it, more than covers it. The thick- 
ness is estimated at 100 miles. 

343. Saturn's ring shines wholly by reflected light derived from 
the sun. This is evident from the fact, that that side only which 
is turned towards the sun is enlightened ; and it is remarkable, 
that the illumination of the ring is greater than that of the planet 
itself, but the outer ring is less bright than the inner. Although, as 
we have already remarked, we view Saturn's ring nearly as though 
we saw it from the sun, yet the plane of the ring produced may 
pass between the earth and the sun, in which case also the ring 
becomes invisible, the illuminated side being wholly turned from 
us. Thus, when the ring is approaching its node at E, a spectator 
at a would have the dark side of the ring presented to him. The 
ring was invisible in 1833, and will be invisible again in 1847. 
At present (1841) it is the northern side of the ring that is seen, 
but in 1855 the southern side will come into view. 

It appears, therefore, that there are three causes for the disap- 
pearance of Saturn's ring ; first, when the edge of the ring is pre- 
sented to the sun ; secondly, when the edge is presented to the 
earth ; and thirdly, when the unilluminated side is towards the 
earth. 

344. Saturn's ring involves in its own plane in about 10£ hours, 
(lOh. 32m. 15 8 .4). La Place inferred this from the doctrine of 
universal gravitation. He proved that such a rotation was neces- 
sary, otherwise the matter of which the ring is composed would 
be precipitated upon its primary. He showed that in order to 
sustain itself, its period of rotation must be equal to the time of 
revolution of a satellite, circulating around Saturn at a distance 
from it equal to that of the middle of the ring, which period would 



SATURN. 203 

be about 10? hours. By means of spots in the ring Dr. Herschel 
followed the ring in its rotation, and actually found its period to 
be the same as assigned by La Place, — a coincidence which beau- 
tifully exemplifies the harmony of truth.* 

345. Although the rings are very nearly concentric, yet recent 
measurements of extreme delicacy have demonstrated, that the 
coincidence is not mathematically exact, but that the center of 
gravity of the rings describes around that of the body a very 
minute orbit. This fact, unimportant as it may seem, is of the 
utmost consequence to the stability of the system of rings. Sup- 
posing them mathematically perfect in their circular form, and 
exactly concentric with the planet, it is demonstrable that they 
would form (in spite of their centrifugal force) a system in a state 
of unstable equilibrium, which the slightest external power would 
subvert — not by causing a rupture in the substance of the rings — 
but by precipitating them unbroken on the surface of the planet. f 
The ring may be supposed of an unequal breadth in its different 
parts, and as consisting of irregular solids, whose common center 
of gravity does not coincide with the center of the figure. Were 
it not for this distribution of matter, its equilibrium would be de- 
stroyed by the slightest force, such as the attraction of a satellite, 
and the ring would finally precipitate itself upon the planet. J 

As the smallest difference of velocity between the planet and 
its rings must infallibly precipitate the rings upon the planet, 
never more to separate, it follows either that their motions in their 
common orbit round the sun, must have been adjusted to each 
other by an external power, with the minutest precision, or that 
the rings must have been formed about the planet while subject 
to their common orbitual motion, and under the full and free in- 
fluence of all the acting forces. 

The rings of Saturn must present a magnificent spectacle from 
those regions of the planet which lie on their enlightened sides, 
appearing as vast arches spanning the sky from horizon to hori- 
zon, and holding an invariable situation among the stars. On 
the other hand, in the region beneath the dark side, a solar eclipse 

* Systeme du Monde, 1. iv. c. 8. t Sir J. Herschel. I La Place. 



204 THE PLANETS. 

of 15 years in duration, under their shadow, must afford (to our 
ideas) an inhospitable abode to animated beings, but ill compen- 
sated by the full light of its satellites. But we shall do wrong 
to judge of the fitness or unfitness of their condition from what 
we see around us, when, perhaps, the very combinations which 
convey to our minds only images of horror, may be in reality 
theatres of the most striking and glorious displays of beneficent 
contrivance.* 

346. Saturn is attended by seven satellites. Although bodies 
of considerable size, their great distance prevents their being vis- 
ible to any telescopes but such as afford a strong light and high 
magnifying powers. The outermost satellite is distant from the 
planet more than 30 times the planet's diameter, and is by far 
the largest of the whole. It is the only one of the series whose 
theory has been investigated further than suffices to verify Kep- 
ler's law of the periodic times, which is found to hold good here 
as well as in the system of Jupiter. It exhibits, like the satellites 
of Jupiter, periodic variations of light, which prove its revolution 
on its axis in the time of a sidereal revolution about Saturn. 
The next satellite in order, proceeding inwards, is tolerably con- 
spicuous ; the three next are very minute, and require pretty pow- 
erful telescopes to see them ; while the two interior satellites, 
which just skirt the edge of the ring, and move exactly in its 
plane, have never been discovered but with the most powerful 
telescopes which human art has yet constructed, and then only 
under peculiar circumstances. At the time of the disappearance 
of the rings (to ordinary telescopes) they were seen by Sir Wil- 
liam Herschel with his great telescope, projected along the edge 
of the ring, and threading like beads the thin fibre of light to which 
the ring is then reduced. Owing to the obliquity of the ring, and 
of the orbits of the satellites to that of their primary, there are no 
eclipses of the satellites, the two interior ones excepted, until near 
the time when the ring is seen edgewise.f 

347. Uranus is the remotest planet belonging to our system, 

\ 

* Sir J. Herschel. t Sir J. Herschel. 



SATURN. 205 

and is rarely visible except to the telescope. Although his diam- 
eter is more than four times that of the earth, (35,112 miles,) yet 
his distance from the sun is likewise nineteen times as great as 
the earth's distance, or about 1 ,800,000,000 miles. His revolution 
around the sun occupies nearly 84 years, so that his position in 
the heavens for several years in succession is nearly stationary. 
His path lies very nearly in the ecliptic, being inclined to it less 
than one degree, (46' 28".44.) 

The sun himself when seen from Uranus dwindles almost to a 
star, subtending as it does an angle of only 1' 40" ; so that the 
surface of the sun would appear there 400 times less than it does 
to us. 

This planet was discovered by Sir William Herschel on the 
13th of March, 1781. His attention was attracted to it by the 
largeness of its disk in the telescope ; and finding that it shifted 
its place among the stars, he at first took it for a comet, but soon 
perceived that its orbit was not eccentric like the orbits of comets, 
but nearly circular like those of the planets. It was then recog- 
nized as a new member of the planetary system, a conclusion 
which has been justified by all succeeding observations. 

348. Uranus is attended by six satellites. So minute objects 
are they that they can be seen only by powerful telescopes. In- 
deed the existence of more than two is still considered as some- 
what doubtful.* These two, however, offer remarkable, and in- 
deed quite unexpected and unexampled peculiarities. Contrary 
to the unbroken analogy of the whole planetary system, the planes 
of their orbits are nearly perpendicular to the ecliptic, being inclined 
no less than 78° 58' to that plane, and in these orbits their motions 
are retrograde ; that is, instead of advancing from west to east 
around their primary, as is the case with all the other planets and 
satellites, they move in the opposite direction.! With this excep- 
tion, all the motions of the planets, whether around their own axes, 
or around the sun, are from west to east. 



* A third satellite of Uranus, is said to have been recently seen at Munich. (Jour. 
Franklin Inst, xxiii, 29.) 
t Sir J. Herschel. 



206 THE PLANETS. 

DP THE NEW PLANETS, CERES, PALLAS, JUNO, AND VESTA. 

349. The commencement of the present century was rendered 
memorable in the annals of astronomy, by the discovery of four 
new planets between Mars and Jupiter. Kepler, from some 
analogy which he found to subsist among the distances of the 
planets from the sun, had long before suspected the existence of 
one at this distance ; and his conjecture was rendered more prob- 
able by the discovery of Uranus, which follows the analogy of 
the other planets. So strongly, indeed, were astronomers im- 
pressed with the idea that a planet would be found between Mars 
■and Jupiter, that in the hope of discovering it, an association was 
formed on the continent of Europe of twenty-four observers, who 
divided the sky into as many zones, one of which was allotted to 
each member of the association. The discovery of the first of 
these bodies was however made accidentally by Piazzi, an astron- 
omer of Palermo, on the first of January, 1801. It was shortly 
afterwards lost sight of on account of its proximity to the sun, 
and was not seen again until the close of the year, when it was 
re-discovered in Germany. Piazzi called it Ceres in honor of the 
tutelary goddess of Sicily, and her emblem, the sickle ?, has been 
adopted as its appropriate symbol. 

The difficulty of finding Ceres induced Dr. Olbers, of Bremen, 
to examine with particular care all the small stars that lie near 
her path, as seen from the earth; and while prosecuting these 
observations, in March, 1802, he discovered another similar body, 
very nearly at the same distance from the sun, and resembling the 
former in many other particulars. The discoverer gave to this 
second planet the name of Pallas, choosing for its symbol the 
lance $ , the characteristic of Minerva. 

350. The most surprising circumstance connected with the 
discovery of Pallas, was the existence of two planets at nearly the 
same distance from the sun, and apparently having a common node. 
On account of this singularity, Dr. Olbers was led to conjecture 
that Ceres and Pallas are only fragments of a larger planet, which 
had formerly circulated at the same distance, and been shattered 
by some internal convulsion. La Grange, a mathematician of the 



NEW PLANETS. 207 

first eminence, investigated the forces that would be necessary to 
detach a fragment from a planet with a velocity that would cause 
it to describe such orbits as these bodies are found to have. The 
hypothesis suggested the probability that there might be other 
fragments, whose orbits, however they might differ in eccentricity 
and inclination, might be expected to cross the ecliptic at a com- 
mon point, or to have the same node. Dr. Olbers, therefore, pro- 
posed to examine carefully every month the two opposite parts 
of the heavens in which the orbits of Ceres and Pallas intersect 
one another, with a view to the discovery of other planets, which 
might be sought for in those parts with greater chance of success 
than in a wider zone, embracing the entire limits of these orbits. 
Accordingly, in 1804, near one of the nodes of Ceres and Pallas, 
a third planet was discovered. This was called Juno, and the 
character § was adopted for its symbol, representing the starry 
sceptre of the queen of Olympus. Pursuing the same researches, 
in 1807, a fourth planet was discovered, to which was given the 
name of Vesta, and for its symbol the character fi was chosen, 
an altar surmounted with a censer holding the sacred fire. 

After this historical sketch, it will be sufficient to classify under 
a few heads the most interesting particulars relating to the New 
Planets. 

351. The average distance of these bodies from the sun is 
261,000,000 miles ; and it is remarkable that their orbits are very 
near together. Taking the distance of the earth from the sun for 
unity, their respective distances are 2.77, 2.77, 2.67, 2.37. 

As they are found to be governed, like the other members of 
the solar system, by Kepler s law, that regulates the distances and 
times of revolution, their periodical times are of course pretty 
nearly equal, averaging about 4? years. 

In respect to the inclination of their orbits, there is considerable 
diversity. The orbit of Vesta is inclined to the ecliptic only 
about 7°, while that of Pallas is more than 34°. They all there- 
fore have a higher inclination than the orbits of the old planets, 
and of course make excursions from the ecliptic beyond the limits 
of the Zodiac. 

The eccentricity of their orbits is also, in general, greater than 



208 THE PLANETS. 

that of the old planets ; and the eccentricities of the orbits of Pal- 
las and Juno exceed that of the orbit of Mercury. 

Their small size constitutes one of their most remarkable pecu- 
liarities. The difficulty of estimating the apparent diameter of 
bodies at once so very small and so far off, would lead us to ex- 
pect different results in the actual estimates. Accordingly, while 
Dr. Herschel estimates the diameter of Pallas at only 80 miles, 
Schroeter places it as high as 2,000 miles, or about the size 
of the moon. The volume of Vesta is estimated at only one fif- 
teen thousandth part of the earth's, and her surface is only about 
equal to that of the kingdom of Spain.* These little bodies are 
surrounded by atmospheres of great extent, some of which are un- 
commonly luminous, and others appear to consist of nebulous mat- 
ter. These planets in general shine with a more vivid light than 
might be expected from their great distance and diminutive size, 



CHAPTER XI. 

MOTIONS OF THE PLANETARY SYSTEM. 

352. We have waited until the learner may be supposed to be 
familiar with the contemplation of the heavenly bodies, individu- 
ally, before inviting his attention to a systematic view of the 
planets, and of their motions around the sun. The time has now 
arrived for entering more advantageously upon this subject, than 
could have been done at an earlier period. 

There are two methods of arriving at a knowledge of the mo- 
tions of the heavenly bodies. One is to begin with the apparent, 
and from these to deduce the real motions ; the other is, to begin 
with considering things as they really are in nature, and then to 
inquire why they appear as they do. The latter of these methods 
is by far the more eligible ; it is much easier than the other, and 

* New Encyc. Brit., Art. Astronomy. 



MOTIONS OF THE PLANETARY SYSTEM. 209 

proceeding from the less difficult to that which is more so, from 
motions that are very simple, to such as are complicated, it finally 
puts the learner in possession of the whole machinery of the heav- 
ens. We shall, in the first place, therefore, endeavor to introduce 
the student to an acquaintance with the simplest motions of the 
planetary system, and afterwards to conduct him gradually 
through such as are more complicated and difficult. 

353. Let us first of all endeavor to acquire an adequate idea of 
absolute space, such as existed before the creation of the world. 
We shall find it no easy matter to form a correct notion of infinite 
space ; but let us fix our attention, for some time, upon extension 
alone, devoid of every thing material, without light or life, and 
without bounds. Of such a space we could not predicate the 
ideas of up or down, east, west, north, or south, but all reference 
to our own horizon (which habit is the most difficult of all to 
eradicate from the mind) must be completely set aside. Into such 
a void we would introduce the Sun. We would contemplate this 
body alone, in the midst of boundless space, and continue to fix 
the attention upon this object, until we had fully settled its rela- 
tions to the surrounding void. The ideas of up and down would 
now present themselves, but as yet there would be nothing to sug- 
gest any notion of the cardinal points. We suppose ourselves 
next to be placed on the surface of the sun, and the firmament of 
stars to be lighted up. The slow revolution of the sun on his axis, 
would be indicated by a corresponding movement of the stars in 
the opposite direction ; and in a period equal to more than 27 of 
our days, the spectator would see the heavens perform a complete 
revolution around the sun, as he now sees them revolve around 
the earth once in 24 hours. The point of the firmament where 
no motion appeared, would indicate the position of one of the 
poles, which being called North, the other cardinal points would 
be immediately suggested. 

Thus prepared, we may now enter upon the consideration of 
the planetary motions. 

354. Standing on the sun, we see all the planets moving slowly 
around the celestial sphere, nearly in the same great high way, and 

27 



210 THE PLANETS. 

in the same direction from west to east. They move, however, 
with very unequal velocities. Mercury makes very perceptible 
progress from night to night, like the moon revolving about the 
earth, his daily progress eastward being about one third as great as 
that of the moon, since he completes his entire revolution in about 
three months. If we watch the course of this planet from night 
to night, we observe it, in its revolution, to cross the ecliptic in 
two opposite points of the heavens, and wander about 7° from 
that plane at its greatest distance from it. Knowing the position 
of the orbit of Mercury with respect to the ecliptic, we may now, 
in imagination, represent that orbit by a great circle passing 
through the center of the planet and the center of the sun, and 
cutting the plane of the ecliptic in two opposite points at an angle 
of 7°. We may imagine the intersection of these two great cir- 
cles, with the celestial vault to be marked out in plain and palpable 
lines on the face of the sky ; but we must bear in mind that these 
orbits are mere mathematical planes, having no permanent exist- 
ence in nature, any more than the path of an eagle flying through 
the sky ; and if we conceive of their orbits as marked on the ce- 
lestial vault, we must be careful to attach to the representation 
the same notion as to a thread or wire carried round to trace out 
the course pursued by a horse in a race-ground.* 

The planes of both the ecliptic and the orbit of Mercury, may 
be conceived of as indefinitely extended to a great distance until 
they meet the sphere of the stars ; but the lines which the earth 
and Mercury describe in those planes, that is, their orbits, may be 
conceived of as comparatively near to the sun. Could we now for 
a moment be permitted to imagine that the planes of the ecliptic, 
and of the orbit of Mercury, were made of thin plates of glass, 
and that the paths of the respective planets were marked out on 
their planes in distinct lines, we should perceive the orbit of the 
earth to be almost a perfect circle, while that of Mercury would 



* It would seem superfluous to caution the reader on so plain a point, did not the 
experience of the instructor constantly show that young learners, from the habit of 
seeing the celestial motions represented in orreries and diagrams, almost always fall 
into the absurd notion of considering the orbits of the planets as having a distinct and 
independent existence. 



MOTIONS OF THE PLANETARY SYSTEM. 211 

appear distinctly elliptical. But having once made use of a palpa- 
ble surface and visible lines to aid us in giving position and figure 
to the planetary orbits, let us now throw aside these devices, and 
hereafter conceive of these planes and orbits as they are in nature, 
and learn to refer a body to a mere mathematical plane, and to 
trace its path in that plane through absolute space. 

355. A clear understanding of the motions of Mercury and of 
the relation of its orbit to the plane of the ecliptic, will render it 
easy to understand the same particulars in regard to each of the 
other planets. Standing on the sun we should see each of the 
planets pursuing a similar course to that of Mercury, all moving 
from west to east, with motions differing from each other chiefly 
in two respects, namely, in their velocities, and in the distances 
to which they ever recede from the ecliptic. 

The earth revolves about the sun very much like Venus, and to 
a spectator on the sun, the motions of these two planets would 
exhibit much the same appearances. We have supposed the ob- 
server to select the plane of the earth's orbit as his standard of 
reference, and to see how each of the other orbits is related to it ; 
but such a selection of the ecliptic is entirely arbitrary ; the spec- 
tator on the sun, who views the motions of the planets as they 
actually exist in nature, would make no such distinction between 
the different orbits, but merely inquire how they were mutually 
related to each other. Taking, however, the ecliptic as the plane 
to which all the others are referred, we do not, as in the case of 
the other planets, inquire how its plane is inclined, nor what are 
its nodes, since it has neither inclination nor node. 

356. Such, in general, are the real motions of the planets, and 
such the appearances which the planetary system would exhibit 
to a spectator at the center of motion. But in order to represent 
correctly the positions of the planetary orbits, at any given time, 
three things must be regarded, — the Inclination of the orbit to the 
ecliptic — the position of the line of the Nodes — and the position of 
the line of the Apsides. In our common diagrams, the orbits are 
incorrectly represented, being all in the same plane, as in the fol- 
lowing diagram, where AEB (Fig. 65,) represents the orbit of 



212 



THE PLANETS. 

Fig. 65. 




Mercury as lying in the same plane with the ecliptic. To exhibit 
its position justly (AB being taken as the line of the nodes) it 
should be elevated on one side about 7° and depressed by the 
same number of degrees on the other side, turning on the line 
AB as on a hinge. But even then the representation may be 
incorrect in other respects, for we have taken it for granted that 
the line of the nodes coincides with the line of the apsides, or 
that the orbit of Mercury cuts the ecliptic in the line AB. Whereas, 
it may lie in any given position with respect to the line of the ap- 
sides depending on the longitude of the nodes. If, for example, the 
line of the nodes had chanced to pass through Taurus and Scor- 
pio instead of Cancer and Capricorn, then it would have been repre- 
sented by the line 8 fll instead of SB VS, and the plane when elevated 
or depressed with respect to the plane of the equator, would be 
turned on this line in our figure.* Moreover, our diagram repre- 
sents the line of the apsides as passing through Cancer and Cap- 
ricorn, whereas it may have any other position among the signs, 
according to the longitudes of the perigee and apogee. 



* The learner will find it useful to construct such representations of the mutual re- 
lations of the planetary orbits of paste board. 



MOTIONS OF THE PLANETARY SYSTEM. 213 

357. The attempt to exhibit the motions of the solar system, and 
the positions of the planetary orbits by means of diagrams, or 
even orreries, is usually a failure. The student who relies exclu- 
sively on such aids as these, will acquire ideas on this subject that 
are both inadequate and erroneous. They may aid reflection, but 
can never supply its place. The impossibility of representing 
things in their just proportions will be evident when we reflect, 
that to do this, if, in an orrery, we make Mercury as large as a 
cherry, we should require to represent the sun by a globe six feet 
in diameter. If we preserve the same proportions in regard to 
distance, we must place Mercury 250 feet, and Uranus 12,500 
feet, or more than two miles from the sun. The mind of the stu- 
dent of astronomy must, therefore, raise itself from such imperfect 
representations of celestial phenomena as are afforded by artificial 
mechanism, and, transferring his contemplations to the celestial 
regions themselves, he must conceive of the sun and planets as 
bodies that bear an insignificant ratio to the immense spaces in 
which they circulate, resembling more a few little birds flying in 
the open sky, than they do the crowded machinery of an orrery. 

358. Having acquired as correct an idea as we are able of the 
planetary system, and of the positions of the orbits with respect to 
the ecliptic, let us next inquire into the nature and causes of the 
apparent motions. 

The apparent motions of the planets are exceedingly unlike the 
real motions, a fact which is owing to two causes ; first, we view 
them out of the center of their orbits ; secondly, we are ourselves in 
motion. From the first cause, the apparent places of the planets 
are greatly changed by perspective ; and from the second cause, 
we attribute to the planets changes of place which arise from our 
own motions of which we are unconscious. 

359. The situation of a heavenly body as seen from the center 
of the sun, is called its heliocentric place ; as seen from the center 
of the earth, its geocentric place. The geocentric motions of the 
planets must, according to what has just been said, be far more 
irregular and complicated than the heliocentric, as will be evident 
from the following diagram, which represents the geocentric mo- 



214 



THE PLANETS. 



tions of Mercury for two entire revolutions, embracing a period 
of nearly six months. 

Let S (Fig. 66,) represent the sun, 1, 2,3, &c. the orbit of Mer- 
cury, a, b, c, &c. that of the earth, and GT the concave sphere of 
the heavens. The orbit of Mercury is divided into 12 equal parts, 
each of which he describes in 7| days, and a portion of the earth's 

Fij 




orbit described by that body in the time that Mercury describes 
the two complete revolutions, is divided into 24 equal parts. Let 
us now suppose that Mercury is at the point 1 in his orbit, when 
the earth is at the point a ; Mercury will then appear in the heav- 
ens at A. In 7^ days Mercury will have reached 2, while the 
earth has reached b, when Mercury will appear at B. By laying 
a ruler on the point c and 3, d and 4, and so on, in the order of the 



MOTIONS OF THE PLANETARY SYSTEM. 215 

alphabet, the successive apparent places of Mercury in the heavens 
will be obtained. 

From A to C, the apparent motion is direct, or in the order of 
the signs ; from C to G it is retrograde ; at G it is stationary 
awhile, and then direct through the whole arc GT. At T the 
planet is again stationary, and afterwards retrograde along the 
arc TX. 

360. Venus exhibits a variety of motions similar to those of 
Mercury, except that the changes do not succeed each other so 
rapidly, since her period of revolution approaches much more 
nearly to that of the earth, 

361. The apparent motions of the superior planets, are, like 
those of Mercury and Venus, alternately direct, stationary, and 
retrograde. In this case, however, the earth moves faster than 
the planet, and the planet has its opposition but no inferior con- 
junction, whereas an inferior planet has its inferior conjunction, 
but no opposition. These differences render the apparent motions 
of the superior planets in some respects unlike those of Mercury 
and Venus. When a superior planet is in conjunction, its motion 
is direct, because, as in the case of Venus in her superior conjunc- 
tion, (see Fig. 60,) the only effect of the earth's motion is to ac- 
celerate it : but when the planet is in opposition, the earth is 
moving past it with a greater velocity, and makes the planet seem 
to move backwards, like the apparent backward motion of a ves- 
sel when we overtake it and pass rapidly by it in a steamboat. 

362. But the various motions of a superior planet will be best 
understood from a diagram. Hence, let S (Fig. 67,) be the sun ; 
B, C, D, E, the orbit of the earth ; b, c, d, &c. the orbit of a supe- 
rior planet, as Jupiter for example ; and IE' a portion of the con- 
cave sphere of the heavens. Let bm be the arc described by Ju- 
piter in the time the earth describes the arc BM ; let be, cd, and 
de, &c. be described by Jupiter while the earth describes BC, 
CD, and DE. Now when the earth is at B and Jupiter at b, he 
will appear in the heavens at B'. When the earth reaches C, the 
planet reaches c and will be seen at C', his motions having been 



216 



THE PLANETS. 

Fig. 67. 




direct from west to east. While the earth moves from C to D 
and from D to E, Jupiter has moved from c to d, and from d to 
e, and will appear to have advanced among the stars from C to 
D', and from D' to E', his motion being still direct, but slower 
than before, as he has passed over only the space D'E' in the 
same time that he before moved through the greater spaces B'C' 
and CD'. 

During the motion of the earth from E to F, and of Jupiter 
from e to f, the earth passes by Jupiter ; and not being conscious 
of our own motion, Jupiter seems to us to have moved backward 
from E' to P. At E' where the direct motion was changed to a 
retrograde, he would appear to be stationary. Upon the arrival 
of the earth at G, and of Jupiter at g, in opposition to the sun, Ju- 
piter will appear at G', having moved with apparently great ve- 
locity over a large space F'G'. While the earth passes from G to 
H, and from H to I, and Jupiter from g to h, and from h to i, he 



DETERMINATION OF THE PLANETARY ORBITS. 217 

will appear to have moved from G' to I'. At I' he will again 
appear stationary in the heavens ; but when he advances from i to 
k in the time the earth moves from I to K, he has described the 
arch I'K', and has therefore resumed his direct motion from west 
to east. While the earth moves from K to L and from L to M, 
and Jupiter through the corresponding spaces Id and 1m, the planet 
will appear still to continue his direct motion from K' to L/ and 
from L' to M' in the heavens. 

Thus, during a period of six months, while the earth is perform- 
ing one half of her annual circuit, Jupiter has a diversity of mo- 
tions, all performed within a small portion of the heavens. 



CHAPTER XII. 

DETERMINATION OF THE PLANETARY ORBITS KEPLER's DISCOVER- 
IES ELEMENTS OF THE ORBIT OF A PLANET QUANTITY OF 

MATTER IN THE SUN AND PLANETS STABILITY OF THE SOLAR 

SYSTEM. 

363. In chapter II, we have shown that the figure of the earth's 
orbit is an ellipse, having the sun in one of its foci, and that the 
earth's radius vector describes equal areas in equal times ; and in 
Chapter III, we have remarked that these are only particular 
examples under the law of Universal Gravitation, as is also the 
additional fact, that the squares of the periodical times of the 
planets are as the cubes of the major axes of their orbits. We 
may now learn, more particularly, the process by which the illus- 
trious Kepler was conducted to the discovery of these grand laws 
of the planetary system. 

364. Ptolemy, while he held that the orbits of the planets were 
perfect circles in which the planets revolved uniformly about the 
earth, was nevertheless obliged to suppose that the earth was 
situated out of the center of the circles, and that at the same 

28 



218 THE PLANETS. 

distance on the other side of the center was situated the point 
(punctum cequans) about which the angular motion of the body 
was equable and uniform. On nearly the same suppositions, Ty- 
cho Brahe had made a great number of very accurate observations 
on the planetary motions, which served Kepler as standards of 
comparison for results, which he deduced from calculations founded 
on the application of geometrical reasoning to hypotheses of his own. 
Kepler first applied himself to investigate the orbit of Mars, the 
motions of which planet appeared more irregular than those of 
any other, except Mercury, which, being seldom seen, had then 
been very little studied. According to the views of Ptolemy and 
Tycho, he at first supposed the orbit to be circular, and the planet 
to move uniformly about a point at a certain distance from the 
sun. He made seventy suppositions before he obtained one that 
agreed with observation, the calculation of which was extremely 
long and tedious, occupying him more than five years.* The sup- 
position of an equable motion in a circle, however varied, could 
not be made to conform to the observations of Tycho, whereas the 
supposition that the orbit was of an oval figure, depressed at the 
sides, but coinciding with a circle at the perihelion, agreed very 
nearly with observation. Such a figure naturally suggested the 
idea of an ellipse, and reasoning on the known properties 
of the ellipse, and comparing the results of calculation with 
actual observation, the agreement was such as to leave no doubt 
that the orbit of Mars is an ellipse, having the sun in one of the 
foci. He immediately conjectured that the same is true of the 
orbits of all the other planets, and a similar comparison of this 
hypothesis with observation, confirmed its truth. Hence he 
established the first great law, that the planets revolve about the 
sun in ellipses, having the sun in one of the foci. 

365. Kepler also discovered from observation, that the velocities 
of the planets when in their apsides, are inversely as their dis- 
tances from the sun, whence it follows that they describe, in these 



* Si te hujus laboriosre methodi pertaesum fuerit, jure mei te misereat, qui earn ad 
minimum septuagies ivi cum plurima temporis jactura ; et mirari desines hunc quintum 
jam annum abire, ex quo Martem aggressus sum. 



kepler's discoveries. 219 

points, equal areas about the sun in equal times. Although he 
could not prove, from observation, that the same was true in 
every point of the orbit, yet he had no doubt that it was so. 
Therefore, assuming this principle as true, and hence deducing the 
equation of the center, (Art. 200,) he found the result to agree 
with observation, and therefore concluded in general, that the 
planets describe about the sun equal areas in equal times. 

366. Having, in his researches that led to the discovery of the 
first of the above laws, found the relative mean distances of the 
planets from the sun, and knowing their periodic times, Kepler 
next endeavored to ascertain if there was any relation between 
them, having a strong passion for finding analogies in nature. 
He saw that the more distant a planet was from the sun, the 
slower it moved ; so that the periodic times of the more distant 
planets would be increased on two accounts, first, because they 
move over a greater space, and secondly, because their motions in 
their orbits are actually slower than the motions of the planets 
nearer the sun. Saturn, for example, is 9£ times further from the 
sun than the earth is, and the circle described by Saturn is greater 
than that of the earth in the same ratio ; and since the earth re- 
volves around the sun in one year, were their velocities equal, the 
periodic time of Saturn would be 9£ years, whereas it is nearly 30 
years. Hence it was evident, that the periodic times of the plan- 
ets increase in a greater ratio than their distances, but in a less 
ratio than the squares of their distances, for on that supposition 
the periodic time of Saturn would be about 90£ years. Kepler 
then took the squares of the times and compared them with the 
cubes of the distances, and found an exact agreement between 
them. Thus he discovered the famous law, that the squares of the 
periodic times of all the planets, are as the cubes of their mean dis- 
tances from the sun.* 

This law is strictly true only in relation to planets whose quan- 
tity of matter in comparison with that of the central body is 
inappreciable. When this is not the case, the periodic time is 
shortened in the ratio of the square root of the sun's mass divi- 

* Vince'a Complete System, I, 98. 



220 THE PLANETS. 

ded by the sun's plus the planet's mass /— r. The mass of 

r \M+mJ 

most of the planets is so small compared with the sun's, that this 
modification of the law is unnecessary except where extreme ac- 
curacy is required. 



ELEMENTS OF THE PLANETARY ORBITS. 

367. The particulars necessary to be known in order to deter- 
mine the precise situation of a planet at any instant, are called 
the Elements of its Orbit, They are seven in number, of which 
the first two determine the absolute situation of the orbit, and the 
other five relate to the motion of the planet in its orbit. These 
elements are, 

(1.) The position of the line of the nodes, 

(2.) The inclination to the ecliptic. 

(3.) The periodic time. 

(4.) The mean distance from the sun, or semi-axis major. 

(5.) The eccentricity. 

(6.) The place of the perihelion. 

(7.) The place of the planet in its orbit at a particular epoch. 

368. It may at first view be supposed that we can proceed to 
find the elements of the orbit of a planet in the same manner as 
we did those of the solar or lunar orbit, namely, by observations 
on the right ascension and declination of the body, converted into 
latitudes and longitudes by means of spherical trigonometry, (See 
Art. 132.) But in the case of the moon, we are situated in the 
center of her motions, and the apparent coincide with the real 
motions ; and, in respect to the sun, our observations on his appa- 
rent motions give us the earth's real motions, allowing 180° differ- 
ence in longitude. But as we have already seen, the motions of 
the planets appear exceedingly different to us, from what they 
would if seen from the center of their motions. It is necessary 
therefore to deduce from observations made on the earth the cor- 
responding results as they would be if viewed from the center of 
the sun ; that is, in the language of astronomers, having the geo- 
centric place of a planet, it is required to find its heliocentric place. 



ELEMENTS OF THE PLANETARY ORBITS. 221 

369. The first steps in this process are the same as in the case of 
the sun and moon. That is, for the purpose of finding the right 
ascension and declination, the planet is observed on the meridian 
with the Transit Instrument and Mural circle, (See Arts. 155 and 
230,) and from these observations, the planet's geocentric longitude 
and latitude are computed by spherical trigonometry. The distance 
of the planet from the sun is known nearly by Kepler's law. From 
these data it is required to find the heliocentric longitude and lati- 
tude. 

Let S and E (Fig. 68,) be the sun and earth, P the planet, PO 
a line drawn from P perpendicular to the ecliptic, SA the direction 
of Aries, and EH parallel to SA, and therefore (on account of 
the immense distance of the fixed stars) also in the direction of 
Aries. Then OEH, being the apparent distance of the planet 
from Aries in the direction of the ecliptic, is the geocentric longi- 
tude, and OEP, being the apparent distance of the planet from the 
ecliptic taken on a secondary to the ecliptic, is the geocentric 
latitude. It is obvious also that the angles OSA and PSO are 



Fig. 68. 




the heliocentric longitude and latitude. The planet's angular dis- 
tance from the sun, PES, is also known from observation. Hence, 
in the triangle SEP, we know SP and SE and the angle SEP, from 
which we can find PE ; and knowing PE and the angle PEO, we 
can find OE, since OEP is a right angled triangle. Hence in the 
triangle SEO, ES and EO, and the angle SEO (=OEH-SEH= 
difference of longitude of the planet and the sun) are known, and 
hence we can obtain OSE, (Art. 135,) which added to the sun's lon- 
gitude ESA, gives us OSA the planet's heliocentric longitude. 



222 THE PLANETS. 

Also, because PS : Rad. :: OP : Sin. PSO 

.-. PSxSin. PSO=OPxRad. 
But EP : Rad. : : OP ; Sin. OEP 

.-. EPxSin. OEP=OPxRad. 

.-. PSxSin. PSO-EPxSin. OEP 

/. PS : EP :: Sin. OEP : Sin. PSO. 
The first three terms of this proposition being known, the last 
is found, which is the heliocentric latitude.* 

370. Having now learned how observations made at the earth 
may be converted into corresponding observations made at the 
sun, we may proceed to explain the mode of finding the several 
elements before enumerated ; although our limits will not permit 
us to enter further into detail on this subject, than to explain the 
leading principles on which each of these elements is determined. f 

371. First, to determine the position of the Nodes, and the In- 
clination of the Orbit. 

These two elements, which determine the situation of the orbit, 
(Art. 367,) may be derived from two heliocentric longitudes and 
latitudes. Let AR and AS (Fig. 69,) Fig. 69. 

be two heliocentric longitudes, PR and 
QS the heliocentric latitudes, and N 
the ascending node. Then, by Napier's 
theorem, (Art. 132,) 
Sin. NR (=AR-AN) =cot pNR: 

tan. PR tan. QS 

Sin. ARxcos. AN — cos. ARxsin. AN J 
'"" tanTPR : 

sin. ASxcos. AN— cos. ASxsin. AN 




But tan. AN: 



tan. QS 
sin. AN _Si n. ARxtan. QS — sin. ASxtan. PR 
cos. AN _ Cos. ARxtan. QS-cos. ASxtan. PR* 



* Brinkley's Elements of Astronomy, p. 164. 

t Most of these elements admit of being determined in several different ways, an 
explanation of which may be found in the larger works on Astronomy, as Vince's Com- 
plete System, Vol. 1. Gregory's Ast. p. 212. Woodhouse, p. 562. 

X Day's Trig. Art. 208. 



ELEMENTS OP THE PLANETARY ORBITS. 223 

But AN is the longitude of the ascending node ; and its value 
is found in terms of the heliocentric longitudes and latitudes pre- 
viously determined, (Art. 369.) 

Again, since AN is found, we may deduce from the first equa- 
tion above the value of PNR, which is the inclination of the orbit.* 

372. Secondly, to find the Periodic Time. 

This element is learned, by marking the interval that passes 
from the time when a planet is in one of the nodes until it returns 
to the same node. We may know when a planet is at the node 
because then its latitude is nothing. If, from a series of observa- 
tions on the right ascension and declination of a planet, we deduce 
the latitudes, and find that one of the observations gives the lati- 
tude 0, we infer that the planet was at that moment at the node. 
But if, as commonly happens, no observation gives exactly 0, then 
we take two latitudes that are nearest to 0, but on opposite sides 
of the ecliptic, one south and the other north, and as the sum of the 
arcs of latitude is to the whole interval, so is one of the arcs to the 
corresponding time in which it was described, which time being 
added to the first observation, or subtracted from the second, will 
give the precise moment when the plane't was at the node. 

By repeated observations it is found, that the nodes of the planets 
have a very slow retrograde motion. 

373. If the orbit of a planet cut the ecliptic at right angles, then 
small changes of place would be attended by appreciable differ- 
ences of latitude ; but in fact the planetary orbits are in general 
but little inclined to the ecliptic, and some of them lie almost in 
the same plane with it. Hence arises a difficulty in ascertaining 
the exact time when a planet reaches its node. Among the most 
valuable observations for determining the elements of a planet's 
orbit, are those made when a superior planet is in or near its oppo- 
sition to the sun, for then the heliocentric and geocentric longitudes 
are the same. When a number of oppositions are observed, the 
planet's motion in longitude as would be observed from the sun will 
be known. The inferior planets also, when in superior conjunction, 

* Brinkley, p. 166. 



224 THE PLANETS. 

have their geocentric and heliocentric longitudes the same. When 
in inferior conjunction, these longitudes differ 180° ; but the in- 
ferior planets can seldom be observed in superior conjunction, on 
account of their proximity to the sun, nor in inferior conjunction 
except in their transits, which occur too rarely to admit of obser- 
vations sufficiently numerous. Therefore, we cannot so readily 
ascertain by simple observation, the motions of the inferior planets 
seen from the sun, as we can those of the superior.* 

Hence, in order to obtain accurately the periodic time of a 
planet, we find the interval elapsed between two oppositions sep- 
arated by a long interval, when the planet was nearly in the same 
part of the Zodiac. From the periodic time, as determined ap- 
proximately by other methods, it may be found when the planet 
has the same heliocentric longitude as at the first observation. 
Hence the time of a complete number of revolutions will be 
known, and thence the time of one revolution. The greater the 
interval of time between the two oppositions, the more accurately 
the periodic time will be obtained, because the errors of observa- 
tion will be divided between a great number of periods ; there- 
fore by using very accurate observations, much precision may be 
attained. For example, the planet Saturn was observed in the 
year 228 B. C. March 2, (according to our reckoning of time,) to 
be near a certain star called y Virginis, and it was at the same 
time nearly in opposition to the sun. The same planet was again 
observed in opposition to the sun, and having nearly the same 
longitude, in Feb. 1714. The exact difference between these dates 
was 1943y. 118d. 21h. 15m. It is known from other sources, that 
the time of a revolution is 29£ years nearly, and hence it was 
found that in the above period there were 66 revolutions of Saturn ; 
and dividing the interval by this number, we obtain 29.444 years, 
which is nearly the periodic time of Saturn according to the most 
accurate determination. 

374. Thirdly, to determine the distance from the sun, and major 
axes of the planetary orbits. 

The distance of the earth from the sun being known, the mean 

* Brinkley, p. 167. 



ELEMENTS OF THE PLANETARY ORBITS. 225 

distance of any planet (its periodic time being known) may be 
found by Kepler's law, that the squares of the periodic times are 
as the cubes of the distances. The method of finding the dis- 
tance of an inferior planet from the sun by observations at the 
greatest elongation, has been already explained, (See Art. 308.) 
The distance of a superior planet may be found from observations 
on its retrograde motion at the time of opposition. The periodic 
times of two planets being known, we of course know their mean 
angular velocities, which are inversely as the times. Therefore, 
let Ee (Fig. 70,) be a very small portion of the earth's orbit, and 
Mm a corresponding portion of that of a superior planet, described 
on the day of opposition^ about the sun S, on which day the three 
bodies lie in one straight line SEMX. Then the angle ESe and 
MSm, representing the respective angular velocities of the two 

Fig. 70. 



bodies are known. Now if em be joined, and prolonged to meet 
SM continued in X, the angle EXe, which is equal to the alternate 
angle Xez/, being equal to the retrogradation of the planet in the 
same time (being known from observation) is also given. Ee, 
therefore, and the angle EXe being given in the right angled tri- 
angle EXe, the side EX is easily calculated, and thus SX becomes 
known. Consequently, in the triangle SwiX, we have given the 
side SX, and the two angles ??iSX and ;?*XS, whence the other 
sides Sm and mX are easily determined. Now Sm is the radius of 
the orbit of the superior planet required, which in this calculation 
is supposed circular as well as that of the earth, — a supposition not 
exact, but sufficiently so to afford a satisfactory approximation 
to the dimensions of its orbit, and which, if the process be often 
repeated, in every variety of situation at which the opposition can 
occur, will ultimately afford an average or mean value of its dis- 
tance fully to be depended on.* 

375. The transverse or major axes of the planetary orbits remain 

* Sir J. Herschel. 

29 



226 



THE PLANETS, 



always the same. Amidst all the perturbations to which other ele- 
ments of the orbit are subject, the line of the apsides is of the same 
invariable length. It is no matter in what direction the planet may- 
be moving at that moment. Various circumstances will influence 
the eccentricity and the position of the ellipse, but none of them 
affects its length. 



Fig. 71. 



376. Fourthly, to determine the place of the perihelion — the epoch 
of passing the perihelion— and the eccentricity. 

There are various methods of finding the eccentricity of a 
planet's orbit and the place of the perihelion, and of course the 
position of the line of the apsides. One is derived from the great- 
est equation of the center, (Art. 200.) The greatest equation is the 
greatest difference that occurs between the mean and the true 
motion of a body revolving in an ellipse. It will be necessary 
first to explain the manner in which the greatest equation is found. 

Let AEBF (Fig. 71,) be the orbit of the planet, having the sun 
in the focus at S. In an ellipse, the square root of the product of 
the semi-axes gives the radius of a circle of the same area as the 
ellipse.* Therefore with the center 
S, at the distance SE=n/AKxOK, 
describe the circle CEGF, then will 
the area of this circle be equal to that 
of the ellipse. At the same time that 
a planet departs from A the aphelion, 
a body begins to move with a uniform 
motion from C through the periphery 
CEGF, and performs a whole revolu- 
tion in the same period that the planet 
describes the ellipse ; the motion of 
this body will represent the equal or 
mean motion of the earth, and it will 
describe around S areas or sectors 
of circles which are proportional to the times, and equal to the 
elliptic areas described in the same time by the planet. Let the 
equal motion, or the angle about S proportional to the time, be 




* Day's Mensuration. 



ELEMENTS OF THE PLANETARY ORBITS. ?27 

CSM, and take ASP equal to the sector CSM ; then the place of 
the planet will be P ; MSC will be the mean anomaly, (Art. 200,) 
DSC the true anomaly, and MSD the equation of the center. Since 
the sectors CSM and ASP are equal, and the part CSD is common 
to both, PACD and SDM are equal ; and since the areas of circu- 
lar sectors are proportional to their arcs, the equation of the center 
is greatest when the area ACPD is greatest, that is, at the point 
E where the ellipse and circle intersect one another. For when 
the planet descends further, to R for instance, the equation becomes 
proportional to the difference of the areas ACE and wER, or to 
the area GBRm, V being the situation of the body moving equa- 
bly ; for the sector CSV will be equal to the elliptic area ASR, 
and taking away the common space CERS, then ACE— RE?w=the 
sector VSm=the equation. At the points E and F, where the 
circle and ellipse intersect, the radius vector of the earth and the 
radius of the circle of equable motion are equal, and of course those 
radii then describe equal areas in equal times; hence, when the 
real motion of the earth is equal to the mean motion, the equation 
of the center is greatest.* The mean motion for any given time 
is easily found ; for the periodic time : 360 : : the given time : the 
number of degrees for that time. Observation shows when the 
actual motion of the planet is the same with this. 

377. Now the equation of the center is greatest twice in the 
revolution, on opposite sides of the orbit, as at E and F, which 
points lie at equal distances from the apsides ; and since the whole 
arc EAF or EBF is known from the time occupied in describing 
it, therefore, by bisecting this arc, we find the points A and B, 
the aphelion and perihelion, and consequently the position of the 
line of the apsides. The time of describing the area EBF being 
known, by bisecting this interval, we obtain the moment of passing 
the perihelion, which gives us the place of the planet in its orbit at 
a particular epoch. 

The amount of the greatest equation obviously depends on the 
eccentricity of the orbit, since it arises wholly from the departure 
of the ellipse from the figure of a perfect circle ; hence, the greatest 

* Gregory's Astronomy, p. 197. 



228 THE PLANETS. 

equation affords the means of determining the eccentricity itself. 
In orbits of small eccentricity, as is the case with most of the 
planetary orbits, it is found that the arc which measures the greatest 
equation is very nearly equal to the distance between the foci,* 
which always equals twice the eccentricity, the eccentricity being 
the distance from the center to the focus. Consequently, 57° 17' 
44".8f : rad. : : half the greatest equation : the eccentricity. 

The foregoing explanations of the methods of finding the ele- 
ments of the orbits, will serve in general to show the learner how 
these particulars are or may be ascertained ; yet the methods actu- 
ally employed are usually more refined and intricate than these. 
In astronomy scarcely an element is presented simple and unmixed 
with others. Its value when first disengaged, must partake of the 
uncertainty to which the other elements are subject ; and can be 
supposed to be settled to a tolerable degree of correctness, only 
after multiplied observations and many revisions.J 

So arduous has been the task of finding the elements of the 
planetary orbits. 

QUANTITY OF MATTER IN THE SUN AND PLANETS. 

378. It would seem at first view very improbable, that an in- 
habitant of this earth would be able to weigh the sun and planets, 
and estimate the exact quantity of matter which they severally con- 
tain. But the principles of Universal Gravitation conduct us to 
this result, by a process remarkable for its simplicity. By com- 
paring the relations of a few elements that are known to us, we 
ascend to the knowledge of such as appeared beyond the pale of 
human investigation. We learn the quantity of matter in a body 
by the force of gravity it exerts. Let us see how this force is ascer- 
tained. 

379. The quantities of matter in two bodies, may be found in 
terms of the distances and periodic times of two bodies revolving 
around them respectively, being as the cubes of the distances divided 
by the squares of the periodic times. 

* Vince's Complete System, I, 113. 

t The value of an arc equal to radius ; for 3.14159 : 1 : : 180 : 57° 17' 44".8. 

t Woodhouse, p. 579. 



QUANTITY OF MATTER IN THE SUN AND PLANETS. 229 

The force of gravity G in a body whose quantity of matter is 
M and distance D, varies directly as the quantity of matter, and 

M 

inversely as the square of the distance ; that is, G ^j^- But it 

is shown by writers on Central Forces, that the force of gravity 
also varies as the distance divided by the square of the periodic 

D M D D 3 

time, or G a—. Therefore, — oc— and M <x— . Thus we may 

find the respective quantities of matter in the earth and the sun, 

by comparing the distance and periodic time of the moon, revolving 

around the earth, with the distance and periodic time of the earth 

revolving around the sun. For the cube of the moon's distance 

from the earth divided by the square of her periodic time, is to the 

cube of the earth's distance from the sun divided by the square of 

her periodic time, as the quantity of matter in the earth is to that 

. . m . 238545 3 95,000,000 3 , _ rt0 „ ™, 

in the sun. That is, -5- : — — ^— ^ : : 1 : 353,385. The most 

27.32 2 365.256 2 

exact determination of this ratio, gives for the mass of the sun 
354,936 times that of the earth. Hence it appears that the sun 
contains more than three hundred and fifty-four thousand times as 
much matter as the earth. Indeed the sun contains eight hundred 
times as much matter as all the planets. 

Another view may be taken of this subject which leads to the 
same result. Knowing the velocity of the earth in its orbit, we 
may calculate its centrifugal force. Now this force is counter- 
balanced, and the earth retained in its orbit, by the attraction of 
the sun, which is proportional to the quantity of matter in the sun. 
Therefore we have only to see what amount of matter is required 
in order to balance the earth's centrifugal force. It is found that 
the earth itself or a body as heavy as the earth acting at the dis- 
tance of the sun, would be wholly incompetent to produce this 
effect, but that in fact it would take more than three hundred and 
fifty-four thousand such bodies to do it. 

380. The mass of each of the other planets that have satellites 
may be found, by comparing the periodic time of one of its satel- 
lites with its own periodic time around the sun. By this means 
we learn the ratio of its quantity of matter to that of the sun. 



THE PLANETS. 

The masses of those planets which have no satellites, as Venus or 
Mars, have been determined, by estimating the force of attraction 
which they exert in disturbing the motions of other bodies. Thus, 
the effect of the moon in raising the tides, leads to a knowledge 
of the quantity of matter in the moon ; and the effect of Venus in 
disturbing the motions of the earth, indicates her quantity of mat- 
ter.* 

381. The quantity of matter in bodies varies as their magnitudes 
and densities conjointly. Hence, their densities vary as their 
masses divided by their magnitudes ; and since we know the mag- 
nitudes of the planets, and can compute as above their masses, we 
can thus learn their densities, which, when reduced to a common 
standard, give us their specific gravities, or show us how much 
heavier they are than water. Worlds therefore are weighed with 
almost as much ease as a pebble, or an article of merchandize. 

The densities and specific gravities of the sun, moon, and planets, 
are estimated as follows :j- 





Density. 


Specific Gravity. 


Sun, . 


0.2543 


1.40 J 


Moon, 


0.6150 


3.37 


Mercury, . 


2.7820 


15.24 


Venus, 


0.9434 


5.17 


Earth, 


1.0000 


5.48 


Mars, ... 


0.1293 


0.71 


Jupiter, . 


0.2589 


1.42 


Saturn, . 


0.1016 


0.56 


Uranus, . 


0.2797 


1.53 



From this table it appears, that the sun consists of matter but 
little heavier than water ; but that the moon is more than three 
times as heavy as water, though less dense than the earth. It also 
appears that the planets near the sun are, as a general fact, more 



* These estimates are made by the most profound investigations in La Place's Me- 
canique Celeste, Vol. III. 

t Francoeur. 

t The earth being taken, according to Bailly, at 5.48, the specific gravities of the 
other bodies (which are found by multiplying the density of each by the specific gravity 
of the earth) are here stated somewhat higher than they are given in most works. 



STABILITY OF THE SOLAR SYSTEM, 231 

dense than those more remote, Mercury being as heavy as the 
heaviest metals except two or three, while Saturn is as light as a 
cork. The decrease of density however is not entirely regular, 
since Venus is a little lighter than the earth, while Jupiter is 
heavier than Mars, and Uranus than Saturn. 

382. The perturbations occasioned in the motions of the planets 
by their action on each other are very numerous, since every body 
in the system exerts an attraction on every other, in conformity 
with the law of Universal Gravitation. Venus and Mars, approach- 
ing as they do at times comparatively near to the earth, sensibly 
disturb its motions, and the satellites of the remote planets greatly 
disturb each other's movements. 



STABILITY OP THE SOLAR SYSTEM. 

383. The derangement which the planets produce in the motion 
of one of their number will be very small in the course of one 
revolution ; but this gives us no security that the derangement may 
not become very large in the course of many revolutions. The 
cause acts perpetually, and it has the whole extent of time to work 
in. Is it not easily conceivable then that in the lapse of ages, the 
derangements of the motions of the planets may accumulate, the 
orbits may change their form, and their mutual distances may be 
much increased or diminished? Is it not possible that these 
changes may go on without limit, and end in the complete subver- 
sion and ruin of the system ? If, for instance, the result of this 
mutual gravitation should be to increase considerably the eccen- 
tricity of the earth's orbit, or to make the moon approach contin- 
ally nearer and nearer to the earth at every revolution, it is easy 
to see that in the one case, our year would change its character, 
producing a far greater irregularity in the distribution of the solar 
heat: in the other, our satellite must fall to the earth, occasioning 
a dreadful catastrophe. If the positions of the planetary orbits 
with respect to that of the earth, were to change much, the plan- 
ets might sometimes come very near us, and thus increase the 
effect of their attraction beyond calculable limits. Under such 
circumstances we might have years of unequal length, and seasons 



232 THE PLANETS. 

of capricious temperature ; planets and moons of portentous size 
and aspect glaring and disappearing at uncertain intervals ; tides 
like deluges sweeping over whole continents ; and, perhaps, the 
collision of two of the planets, and the consequent destruction of 
all organization on both of them. The fact really is, that changes 
are taking place in the motions of the heavenly bodies, which have 
gone on progressively from the first dawn of science. The eccen- 
tricity of the earth's orbit has been diminishing from the earliest 
observations to our times. The moon has been moving quicker 
from the time of the first recorded eclipses, and is now in advance 
by about four times her own breadth, of what her own place 
would have been if it had not been affected by this acceleration. 
The obliquity of the ecliptic also, is in a state of diminution, and is 
now about two fifths of a degree less than it was in the time of 
Aristotle.* 

384. But amid so many seeming causes of irregularity, and ruin, 
it is worthy of grateful notice, that effectual provision is made for 
the stability of the solar system. The full confirmation of this fact ? 
is among the grand results of Physical Astronomy. Newton did 
not undertake to demonstrate either the stability or instability of 
the system. The decision of this point required a great number 
of preparatory steps and simplifications, and such progress in the 
invention and improvement of mathematical methods as occu- 
pied the best mathematicians of Europe for the greater part of 
the last century. Towards the end of that time, it was shown by 
La Grange and La Place, that the arrangements of the solar sys- 
tem are stable ; that, in the long run, the orbits and motions remain 
unchanged ; and that the changes in the orbits, which take place 
in shorter periods, never transgress certain very moderate limits. 
Each orbit undergoes deviations on this side and on that side of its 
average state ; but these deviations are never very great, and it 
finally recovers from them, so that the average is preserved. The 
planets produce perpetual perturbations in each other's motions, 
but these perturbations are not indefinitely progressive, but period- 
ical, reaching a maximum value and then diminishing. The pe- 

* Whewell, in the Bridgewater Treatises, p. 128. 



STABILITY OF THE SOLAR SYSTEM. 233 

riods which this restoration requires are for the most part enor- 
mous — not less than thousands, and in some instances millions of 
years. Indeed some of these apparent derangements, have been 
going on in the same direction from the creation of the world. 
But the restoration is in the sequel as complete as the derange- 
ment ; and in the mean time the disturbance never attains a suf- 
ficient amount seriously to affect the stability of the system.* I 
have succeeded in demonstrating (says La Place) that, whatever be 
the masses of the planets, in consequence of the fact that they all 
move in the same direction, in orbits of small eccentricity, and but 
slightly inclined to each other, their secular irregularities are pe- 
riodical and included within narrow limits ; so that the planetary 
system will only oscillate about a mean state, and will never de- 
viate from it except by a very small quantity. The ellipses of the 
planets have been and always will be nearly circular. The eclip- 
tic will never coincide with the equator ; and the entire extent of 
the variation in its inclination, cannot exceed three degrees. 

385. To these observations of La Place, Professor Whewellf 
adds the following on the importance, to the stability of the solar 
system, of the fact that those planets which have great masses 
have orbits of small eccentricity. The planets Mercury and Mars, 
which have much the largest eccentricity among the old planets, 
are those of which the masses are much the smallest. The mass 
of Jupiter is more than two thousand times that of either of these 
planets. If the orbit of Jupiter were as eccentric as that of Mer- 
cury, all the security for the stability of the system, which analy- 
sis has yet pointed out, would disappear. The earth and the 
smaller planets might in that case change their nearly circular or- 
bits into very long ellipses, and thus might fall into the sun, or fly 
off into remote space. It is further remarkable that in the newly 
discovered planets, of which the orbits are still more eccentric 
than that of Mercury, the masses are still smaller, so that the same 
provision is established in this case also. 

* Whewell, in the Bridgewater Treatises, p. 128. 

t Bridgewater Treatises, p. 131. See also Playfair's Outlines, 2, 290. 

30 



CHAPTER XIII. • 

OF COMETS. 

386. A Comet,* when perfectly formed, consists of three parts, 
the Nucleus, the Envelope, and the Tail. The Nucleus, or body 
of the comet, is generally distinguished by its forming a bright 
point in the center of the head, conveying the idea of a solid, or at 
least of a very dense portion of matter. Though it is usually 
exceedingly small when compared with the other parts of the 
comet, yet it sometimes subtends an angle capable of being meas- 
ured by the telescope. The Envelope, (sometimes called the coma) 
is a dense nebulous covering, which frequently renders the edge 
of the nucleus so indistinct, that it is extremely difficult to ascer- 
tain its diameter with any degree of precision. Many comets have 
no nucleus, but present only a nebulous mass extremely attenuated 
on the confines, but gradually increasing in density towards the 
center. Indeed there is a regular gradation of comets, from such 
as are composed merely of a gaseous or vapory medium, to those 
which have a well defined nucleus. In some instances on record, 
astronomers have detected with their telescopes small stars through 
the densest part of a comet. 

The Tail is regarded as an expansion or prolongation of the 
coma ; and, presenting as it sometimes does, a train of appalling 
magnitude, and of a pale, portentous light, it confers on this class 
of bodies their peculiar celebrity. 

387. The number of comets belonging to the solar system, is 
probably very great. Many, no doubt, escape observation by being 
above the horizon in the day time. Seneca mentions, that during 
a total eclipse of the sun, which happened 60 years before the 
Christian era, a large and splendid comet suddenly made its 



t <c<5/j>7, coma, from the bearded appearance of comets. 



COMETS, 



235 



Fig. 71' 



Fig. 71". 




W:" 










COMET OF 1811. 



COMET OF 16-v). 



appearance, being very near the sun. The elements of at least 130 
have been computed, and arranged in a table for future compari- 
son. Of these six are particularly remarkable, viz. the comets of 
1680, 1770, and 1811 ; and those which bear the names of Halley, 
Biela, and Encke. The comet of 1680, was remarkable not only 
for its astonishing size and splendor, and its near approach to the 
sun, but is celebrated for having submitted itself to the observa- 
tions of Sir Isaac Newton, and for having enjoyed the signal honor 
of being the first comet whose elements were determined on the 
sure basis of mathematics. The comet of 1770, is memorable for 
the changes its orbit has undergone by the action of Jupiter, as 
will be more particularly related in the sequel. The comet of 
1811 was the most remarkable in its appearance of all that have 
been seen in the present century. Halley's comet (the same 
which re-appeared in 1835) is distinguished as that whose return 
was first successfully predicted, and whose orbit was first deter- 
mined ; and Biela's and Encke's comets are well known, for their 



236 COMETS. 

short periods of revolution, which subject them frequently to the 
view of astronomers. 

388. In magnitude and brightness comets exhibit a great diver- 
sity. History informs us of comets so bright as to be distinctly 
visible in the day time, even at noon and in the brightest sunshine. 
Such was the comet seen at Rome a little before the assassination 
of Julius Caesar. The comet of 1680 covered an arc of the 
heavens of 97°, and its length was estimated at 123,000,000 
miles.* That of 1811, had a nucleus of only 428 miles in diame- 
ter, but a tail 132,000,000 miles long.-]- Had it been coiled around 
the earth like a serpent, it would have reached round more than 
5,000 times. Other comets are of exceedingly small dimensions, 
the nucleus being estimated at only 25 miles ; and some which are 
destitute of any perceptible nucleus, appear to the largest tele- 
scopes, even when nearest to us, only as a small speck of fog, or 
as a tuft of down. The majority of comets can be seen only by 
the aid of the telescope. 

The same comet, indeed, has often very different aspects, at its 
different returns. Halley's comet in 1305 was described by the 
historians of that age, as cometa horrendce magnitudinis ; in 1456 
its tail reached from the horizon to the zenith, and inspired such 
terror, that by a decree of the Pope of Rome, public prayers were 
offered up at noon-day in all the Catholic churches to deprecate 
the wrath of heaven, while in 1682, its tail was only 30° in length, 
and in 1759 it was visible only to the telescope, until after it had 
passed its perihelion. At its recent return in 1835, the greatest 
length of the tail was about 12°. J These changes in the appear- 
ances of the same comet are partly owing to the different positions 
of the earth with respect to them, being sometimes much nearer 
to them when they cross its track than at others ; also one specta- 
tor so situated as to see the comet at a higher angle of elevation or 
in a purer sky than another, will see the train longer than it 
appears to another less favorbly situated ; but the extent of the 



* Arago. t Milne's Prize Essay on Comets. 

I But might be seen much longer by indirect vision. (Prof. Joslm, Am. Jour. Sci- 
ence, 31, 328.) 



COMETS. 237 

changes are such as indicate also a real change in their magnitude 
and brightness. 

389. The periods of comets in their revolutions around the sun, 
are equally various. Encke's comet, which has the shortest known 
period, completes its revolution in 3^ years, or more accurately, 
in 1208 days ; while that of 1811 is estimated to have a period of 
3383 years.* 

390. The distances to which different comets recede from the sun, 
are also very various. While Encke's comet performs its entire 
revolution within the orbit of Jupiter, Halley's comet recedes from 
the sun to twice the distance of Uranus, or nearly 3600,000,000 
miles. Some comets, indeed, are thought to go to a much greater 
distance from the sun than this, while some even are supposed to 
pass into parabolic or hyperbolic orbits, and never to return. 

391. Comets shine by reflecting the light of the sun. In one or 
two instances they have exhibited distinct phases,-\ although the 
nebulous matter with which the nucleus is surrounded, would com- 
monly prevent such phases from being distinctly visible, even 
when they would otherwise be apparent. Moreover, certain 
qualities of polarized light enable the optician to decide whether 
the light of a given body is direct or reflected ; and M. Arago, of 
Paris, by experiments of this kind on the light of the comet of 
1819, ascertained it to be reflected light. J 

392. The tail of a comet usually increases very much as it 
approaches the sun ; and frequently does not reach its maximum 
until after the perihelion passage. In receding from the sun, the 
tail again contracts, and nearly or quite disappears before the body 
of the comet is entirely out of sight. The tail is frequently divi- 
ded into two portions, the central parts, in the direction of the 
axis, being less bright than the marginal parts. In 1744, a comet 
appeared which had six tails, spread out like a fan. 

The tails of comets extend in a direct line from the sun, although 
they are usually more or less curved, like a long quill or feather, 

* Milne. t Delambre, t. 3, p. 400. X Francoeur, 181. 



238 COMETS. 

being convex on the side next to the direction in which they 
are moving ; a figure which may result from the less velocity of 
the portions most remote from the sun. Expansions of the Enve- 
lope have also been at times observed on the side next the sun,* 
but these seldom attain any considerable length. 

393. The quantity of matter in comets is exceedingly small. 
Their tails consist of matter of such tenuity that the smallest stars 
are visible through them. They can only be regarded as great 
masses of thin vapor, susceptible of being penetrated through 
their whole substance by the sunbeams, and reflecting them alike 
from their interior parts and from their surfaces. It appears, per- 
haps, incredible that so thin a substance should be visible by re- 
flected light, and some astronomers have held that the matter of 
comets is self-luminous ; but it requires but very little light to ren- 
der an object visible in the night, and a light vapor may be visible 
when illuminated throughout an immense stratum, which could not 
be seen if spread over the face of the sky like a thin cloud. The 
highest clouds that float in our atmosphere, must be looked upon 
as dense and massive bodies, compared with the filmy and all but 
spiritual texture of a comet, f 

394. The small quantity of matter in comets is proved by the 
fact that they have sometimes passed very near to some of the planets 
without disturbing their motions in any appreciable degree. Thus 
the comet of 1770, in its way to the sun, got entangled among the 
satellites of Jupiter, and remained near them four months, yet it did 
not perceptibly change their motions. The same comet also came 
very near the earth ; so near, that, had its mass been equal to that 
of the earth, it would have caused the earth to revolve in an orbit 
so much larger than at present, as to have increased the length of 
the year 2h. 47m. J Yet it produced no sensible effect on the 
length of the year, and therefore its mass, as is shown by La 
Place, could not have exceeded 50V0 °f that of the earth, and 
might have been less than this to any extent. It may indeed be 



* See Dr. Joslin's remarks on Halley's comet, Amer. Jour. Science, Vol. 31. 
t Sir. J. Herschel. X La Place. 



COMETS. 239 

asked, what proof we have that comets have any matter, and are not 
mere reflexions of light. The answer is that, although they are not 
able by their own force of attraction to disturb the motions of the 
planets, yet they are themselves exceedingly disturbed by the action 
of the planets, and in exact conformity with the laws of universal 
gravitation. A delicate compass may be greatly agitated by the 
vicinity of a mass of iron, while the iron is not sensibly affected 
by the attraction of the needle. 

By approaching very near to a large planet, a comet may have 
its orbit entirely changed. This fact is strikingly exemplified in 
the history of the comet of 1770. At its appearance in 1770, 
its orbit was found to be an ellipse, requiring for a complete revo- 
lution only 5s years ; and the wonder was, that it had not been 
seen before, since it was a very large and bright comet. Astron- 
omers suspected that its path had been changed, and that it had 
been recently compelled to move in this short ellipse, by the dis- 
turbing force of Jupiter and his satellites. The French Institute, 
therefore, offered a high prize for the most complete investigation 
of the elements of this comet, taking into account any circum- 
stances which could possibly have produced an alteration in its 
course. By tracing back the movements of this comet for some 
years previous, to 1770, it was found that, at the beginning of 
1767, it had entered considerably within the sphere of Jupiter's 
attraction. Calculating the amount of this attraction from the 
know T n proximity of the two bodies, it was found what must have 
been its orbit previous to the time when it became subject to the 
disturbing action of Jupiter. The result showed that it then 
moved in an ellipse of greater extent, having a period of 50 years, 
and having its perihelion instead of its aphelion near Jupiter. It 
was therefore evident why, as long as it continued to circulate in 
an orbit so far from the center of the system, it w r as never visible 
from the earth. In January, 1767, Jupiter and the comet happened 
to be very near one another, and as both were moving in the same 
direction, and nearly in the same plane, they remained in the 
neighborhood of each other for several months, the planet being 
between the comet and the sun. The consequence was, that the 
comet's orbit was changed into a smaller ellipse, in which its revo- 
lution was accomplished in 5* years. But as it was approaching 



240 COMETS. 

the sun in 1779, it happened again to fall in with Jupiter. It was 
in the month of June, that the attraction of the planet began to 
have a sensible effect ; and it was not until the month of October 
following that they were finally separated. 

At the time of their nearest approach, in August, Jupiter was 
distant from the comet only ¥ ^ T of its distance from the sun, and 
exerted an attraction upon it 225 times greater than that of the 
sun. By reason of this powerful attraction, Jupiter being further 
from the sun than the comet, the latter was drawn out into a new 
orbit, which even at its perihelion came no nearer to the sun than 
the planet Ceres. In this third orbit, the comet requires about 20 
years to accomplish its revolution; and being at so great a dis- 
tance from the earth, it is invisible, and will forever remain so un- 
less, in the course of ages, it may undergo new perturbations, and 
move again in some smaller orbit as before.* 



ORBITS AND MOTIONS OF COMETS. 

395. The planets, as we have seen, (with the exception of the 
four new ones, which seem to be an intermediate class of bodies 
between planets and comets,) move in orbits which are nearly cir- 
cular, and all very near to the plane of the ecliptic, and all move 
in the same direction from west to east. But the orbits of comets 
are far more eccentric than those of the planets ; they are in- 
clined to the ecliptic at various angles, being sometimes even 
nearly perpendicular to it ; and the motions of comets are some- 
times retrograde. 

396. The Elements of a comet are five, viz. (1) The perihelion 
distance ; (2) longitude of the perihelion ; (3) longitude of the node ; 
(4) inclination of the orbit ; (5) time of the perihelion passage. 

The investigation of these elements is a problem extremely in- 
tricate, requiring for its solution, a skilful and laborious applica- 
tion of the most refined analysis. Newton himself, pronounced it 
Problema longe difficilimum ; and with all the advantages of the 
most improved state of science, the determination of a comet's 

* Milne. 



ORBITS AND MOTIONS OF COMETS. 



241 



orbit is considered one of the most complicated problems in as- 
tronomy. This difficulty arises from several circumstances pecu- 
liar to comets. In the first place, from the elongated form of the 
orbits which these bodies describe, it is during only a very small 
portion of their course, that they are visible from the earth, and 
the observations made in that short period, cannot afterwards be 
verified on more convenient occasions ; whereas in the case of the 
planets, whose orbits are nearly circular, and whose movements may 
be followed uninterruptedly throughout a complete revolution, no 
such impediments to the determination of their orbits occur. There 
is also some unavoidable uncertainty in observations made upon 
bodies whose outlines are so ill-defined. In the second place, there 
are many comets which move in a direction opposite to the order 
of the signs in the zodiac, and sometimes nearly perpendicular to 
the plane of the ecliptic ; so that their apparent course through the 
heavens is rendered extremely complicated, on account of the con- 
trary motion of the earth. In the third place, as there may be 
a multitude of elliptic orbits, whose perihelion distances are equal, 
it is obvious that, in the case of very eccentric orbits, the slightest 
change in the position of the curve near the vertex, where alone 
the comet can be observed, must occasion a very sensible differ- 
ence in the length of the orbit (as will be obvious from Fig. IV" ;) 
and therefore, though a small error produces no perceptible dis- 
crepancy between the observed and the calculated course, while 




242 COMETS. 

the comet remains visible from the earth, its effect when diffused 
over the whole extent of the orbit, may acquire a most material or 
even a fatal importance. 

On account of these circumstances, it is found exceedingly diffi- 
cult to lay down the path which a comet actually follows through 
the whole system, and least, of all, possible to ascertain with accu- 
racy, the length of the major axis of the ellipse, and consequently 
the periodical revolution.* An error of only a few seconds may 
cause a difference of many hundred years. In this manner, though 
Bessel determined the revolution of the comet of 1769 to be 2089 
years, it was found that an error of no more than 5" in observation, 
would alter the period either to 2678 years, or to 1692 years. 
Some astronomers, in calculating the orbit of the great comet of 
1680, have found the length of its greater axis 426 times the 
earth's distance from the sun, and consequently its period 8792 
years ; whilst others estimate the greater axis 430 times the comet's 
distance, which alters the period to 8916 years. Newton and 
Halley, however, judged that this comet accomplished its revolu- 
tion in only 570 years. 

397. Disheartened by the difficulty of attaining to any precision 
in that circumstance, by which an elliptic orbit is characterized, 
and, moreover, taking into account the laborious calculations 
necessary for its investigation, astronomers usually satisfy them- 
selves with ascertaining the elements of a comet on the supposition 
of its describing a parabola ; and, as this is a curve whose axis is 
infinite, the procedure is greatly simplified by leaving entirely out 
of consideration the periodic revolution. It is true that a parabola 
may not represent with mathematical strictness the course which 
a comet actually follows ; but as a parabola is the intermediate 
curve between the hyperbola and ellipse, it is found that this 
method, which is so much more convenient for computation, also 
accords sufficiently with observations, except in cases when the 
ellipse is a comparatively short one, as that of Encke's comet, for 
example. 

* For when we know the length of the major axis, we can find the periodic time by 
Kepler's law, which applies as well to comets as to planets. 



ORBITS AND MOTIONS OP COMETS. 



243 



398. The elements of a comet, with the exception of its periodic 
time, are calculated in a manner similar to those of the planets. 
Three good observations on the right ascension and declination of 
the comet (which are usually found by ascertaining its position 
with respect to certain stars, whose right ascensions and declina- 
tions are accurately known) afford the means of calculating these 
elements. 

The appearance of the same comet at different periods of its 
return are so various, (Art. 388,) that we can never pronounce a 
given comet to be the same with one that has appeared before, 
from any peculiarities in its physical aspect. The identity of a 
comet with one already on record, is determined by the identity 
of the elements. It was by this means that Halley first established 
the identity of the comet which bears his name, with one that 
had appeared at several preceding ages of the world, of which 
so many particulars were left on record, as to enable him to cal- 
culate the elements at each period. These were as in the follow- 
ing table. 



Time of appear. 


Inclin. of the orbit. 


Long, of the Node. 


Long, of Per. 


Per. Diat. 


Course. 


1456 


17° 56' 


48° 30' 


301° 00 


0.58 


Retrograde. 


1531 


17 56 


49 25 


301 39 


0.57 


Retrograde. 


1607 


17 02 


50 21 


302 16 


0.58 


Retrograde. 


1682 


17 42 


50 48 


301 36 


0.58 


Retrograde. 



On comparing these elements, no doubt could be entertained 
that they belonged to one and the same body ; and since the in- 
terval between the successive returns was seen to be 75 or 76 
years, Halley ventured to predict that it would again return in 
1758. Accordingly, the astronomers who lived at that period, 
looked for its return with the greatest interest. It was found 
however, that on its way towards the sun it would pass very near 
to Jupiter and Saturn, and by their action on it, it would be re- 
tarded for a long time. Clairaut, a distinguished French mathe- 
matician, undertook the laborious task of estimating the exact 
amount of this retardation, and found it to be no less than 618 
days, namely, 100 days by the action of Jupiter, and 518 days by 
that of Saturn. This would delay its appearance until early in 
the year 1759, and Clairaut fixed its arrival at the perihelion within 



244 COMETS. 

a month of April 13th. It came to the perihelion on the 12th of 
March. 

399. The return of Halley's comet in 1835, was looked for with 
no less interest than in 1759. Several of the most accurate math- 
ematicians of the age had calculated its elements with inconceiva- 
ble labor. Their zeal was rewarded by the appearance of the 
expected visitant at the time and place assigned ; it traversed the 
northern sky presenting the very appearances, in most respects, 
that had been anticipated ; and came to its perihelion on the 16th 
of November, within two days of the time predicted by Ponte- 
coulant, a French mathematician who had, it appeared, made the 
most, successful calculation.* On its previous return, it was 
deemed an extraordinary achievement to have brought the pre- 
diction within a month of the actual time. 

Many circumstances conspired to render this return of Halley's 
comet an astronomical event of transcendent interest. Of all the 
celestial bodies, its history was the most remarkable ; it afforded 
most triumphant evidence of the truth of the doctrine of univer- 
sal gravitation, and of course of the received laws of astronomy ; 
and it inspired new confidence in the power of that instrument, 
(the Calculus,) by means of which its elements had been investi- 
gated. 

400. Encke's comet, by its frequent returns, affords peculiar fa- 
cilities for ascertaining the laws of its revolution ; and it has kept 
the appointments made for it, with great exactness. On its re- 
turn in 1839 it exhibited to the telescope a globular mass of 
nebulous matter, resembling fog, and moved towards its perihelion 
with great rapidity. 

But what has made Encke's comet particularly famous, is its 
having first revealed to us the existence of a B,esisting Medium in 
the planetary spaces. It has long been a question whether the 
earth and planets revolve in a perfect void, or whether a fluid of 
extreme rarity may not be diffused through space. A perfect 



* See Professor Loomis's Observations on Halley's Comet, Amer. Jour. Science, 30. 
209. 



COMETS. 245 

vacuum was deemed most probable, because no such effects on the 
motions of the planets could be detected as indicated that they en- 
countered a resisting medium. But a feather or a lock of cotton 
propelled with great velocity, might render obvious the resistance 
of a medium which would not be perceptible in the motions of a 
cannon ball. Accordingly, Encke's comet is thought to have plainly 
suffered a retardation from encountering a resisting medium in the 
planetary regions. The effect of this resistance, from the first dis- 
covery of the comet to the present time, has been to diminish the 
time of its revolution about two days. Such a resistance, by de- 
stroying a part of the projectile force, would cause the comet to 
approach nearer to the sun, and thus to have its periodic time 
shortened. The ultimate effect of this cause will be to bring the 
comet nearer to the sun at every revolution, until it finally falls 
into that luminary, although many thousand years will be required 
to produce this catastrophe.* It is conceivable, indeed, that the 
effects of such a resistance may be counteracted by the attraction 
of one or more of the planets near which it may pass in its succes- 
sive returns to the sun. 

401. It is peculiarly interesting to see a portion of matter of a 
tenuity exceeding the thinnest fog, pursuing its path in space, in 
obedience to the same laws as those which regulate such large and 
heavy bodies as Jupiter or Saturn. In a perfect void, a speck of 
fog if propelled by a suitable projectile force would revolve around 
the sun, and hold on its way through the widest orbit, with as sure 
and steady a pace as the heaviest and largest bodies in the system. 

402. Of the physical nature of comets, little is understood. It is 
usual to account for the variations which their tails undergo by 
referring them to the agencies of heat and cold. The intense heat 
to which they are subject in approaching so near the sun as some 
of them do, is alleged as a sufficient reason for the great expansion 
of thin nebulous atmospheres forming their tails ; and the incon- 
ceivable cold to which they are subject in receding to such a dis- 



* See Professor WhewelPs Observations on this subject in Bridgewater Treatises, 
Ch. viii. 



246 COMETS. 

tance from the sun, is supposed to account for the condensation of 
the same matter until it returns to its original dimensions. Thus 
the great comet of 1680 at its perihelion approached 166 times 
nearer the sun than the earth, being only 130,000 miles from the 
surface of the sun.* The heat which it must have received, was 
estimated to be equal to 28,000 times that which the earth receives 
in the same time, and 2000 times hotter than red hot iron. This 
temperature would be sufficient to volatilize the most obdurate 
substances, and to expand the vapor to vast dimensions ; and the 
opposite effects of the extreme cold to which it would be subject 
in the regions remote from the sun, would be adequate to condense 
it into its former volume. 

This explanation however, does not account for the direction 
of the tail, extending as it usually does, only in a line opposite to 
the sun. Some writers therefore, as Delambre, suppose that the 
nebulous matter of the comet after being expanded to such a vol- 
ume, that the particles are no longer attracted to the nucleus un- 
less by the slightest conceivable force, are carried off in a direction 
from the sun, by the impulse of the solar rays themselves.f But 
to assign such a power of communicating motion to the sun's rays 
while they have never been proved to have any momentum, is 
unphilosophical ; and we are compelled to place the phenomena 
of comets' tails among the points of astronomy yet to be ex- 
plained. 

403. Since those comets which have their perihelion very near 
the sun, like the comet of 1680, cross the orbits of all the planets, 
the possibility that one of them may strike the earth, has frequently 
been suggested. Still it may quiet our apprehensions on this sub- 
ject, to reflect on the vast extent of the planetary spaces, in which 
these bodies are not crowded together as we see them erroneously 
represented in orreries and diagrams, but are sparsely scattered at 
immense distances from each other. They are like insects flying 
in the expanse of heaven. If a comet's tail lay with its axis in the 
plane of the ecliptic when it was near the sun, we can imagine that 
the tail might sweep over the earth ; but the tail may be situated 

* See Principia, Lib. in, 41. t Delambre's Astronomy, t. 3, p. 401 



COMETS. 247 

at any angle with the ecliptic as well as in the same plane with it, 
and the chances that it will not be in the same plane, are almost 
infinite. It is also extremely improbable that a comet will cross 
the plane of the ecliptic precisely at the earth's path in that plane, 
since it may as probably cross it at any other point, nearer or 
more remote from the sun. Still some comets have occasionally 
approached near to the earth. Thus Biela's comet in returning to 
the sun in 1832, crossed the ecliptic very near to the earth's track, 
and had the earth been then at that point of its orbit, it might 
have passed through a portion of the nebulous atmosphere of the 
comet. The earth was within a month of reaching that point. 
This might at first view seem to involve some hazard ; yet we must 
consider that a month short implied a distance of nearly 50,000,000 
miles. La Place has assigned the consequences that would ensue 
in case of a direct collision between the earth and a comet ; # but 
terrible as he has represented them on the supposition that the 
nucleus of the comet is a solid body, yet considering a comet (as 
most of them doubtless are) as a mass of exceedingly light nebu- 
lous matter, it is not probable, even were the earth to make its 
way directly through a comet, that a particle of the comet would 
reach the earth. The portions encountered by the earth, would 
be arrested by the atmosphere, and probably inflamed ; and they 
would perhaps exhibit on a more magnificent scale than was ever 
before observed, the phenomena of shooting stars, or meteoric 
showers. 

* Syst. du Monde, 1. iv, c. 4. 



PART III. — OF THE FIXED STARS AND SYSTEM OF THE 
THE WORLD. 



CHAPTER I 



OF THE FIXED STARS CONSTELLATIONS. 

404. The Fixed Stars are so called, because, to common ob- 
servation, they always maintain the same situations with respect 
to one another. 

The stars are classed, by their apparent magnitudes. The whole 
number of magnitudes recorded are sixteen, of which the first six 
only are visible to the naked eye ; the rest are telescopic stars. As 
the stars which are now grouped together under one of the first 
six magnitudes are very unequal among themselves, it has recently 
been proposed to subdivide each class into three, making in all 
eighteen instead of six magnitudes visible to the naked eye. 
These magnitudes are not determined by any very definite scale, 
but are merely ranked according to their relative degrees of 
brightness, and this is left in a great measure to the decision of the 
eye alone, although it would appear easy to measure the compar- 
ative degree of light in a star by a photometer, and upon such 
measurement to ground a more scientific classification of the stars. 
The brightest stars to the number of 15 or 20 are considered as 
stars of the first magnitude ; the 50 or 60 next brightest, of the 
second magnitude ; the next 200 of the third magnitude ; and thus 
the number of each class increases rapidly as we descend the scale, 
so that no less than fifteen or twenty thousand are included within 
the first seven magnitudes. 

405. The stars have been grouped in Constellations from the 
most remote antiquity : a few, as Orion, Bootes, and Ursa Major, 
are mentioned in the most ancient writings under the same names 



FIXED STARS. 249 

as they bear at present. The names of the constellations are 
sometimes founded on a supposed resemblance to the objects to 
which the names belong ; as the Swan and the Scorpion were evi- 
dently so denominated from their likeness to those animals ; but 
in most cases it is impossible for us to find any reason for desig- 
nating a constellation by the figure of the animal or the hero which 
is employed to represent it. These representations were probably 
once blended with the fables of pagan mythology. The same fig- 
ures, absurd as they appear, are still retained for the convenience 
of reference ; since it is easy to find any particular star, by speci- 
fying the part of the figure to which it belongs, as when we say a 
star is in the neck of Taurus, in the knee of Hercules, or in the 
tail of the Great Bear. This method furnishes a general clue to 
its position ; but the stars belonging to any constellation are dis- 
tinguished according to their apparent magnitudes as follows : — 
first, by the Greek letters, Alpha, Beta, Gamma, &c. Thus a 
Ononis, denotes the largest star in Orion, (3 Andromedce, the 
second star in Andromeda, and y Leonis, the third brightest star in 
the Lion. Where the number of the Greek letters is insufficient 
to include all the stars in a constellation, recourse is had to the 
letters of the Roman alphabet, a, b, c, &c. ; and, in cases where 
these are exhausted, the final resort is to numbers. This is evi- 
dently necessary, since the largest constellations contain many 
hundreds or even thousands of stars. Catalogues of particular 
stars have also been published by different astronomers, each 
author numbering the individual stars embraced in his list, accord- 
ing to the places they respectively occupy in the catalogue. 
These references to particular catalogues are sometimes entered 
on large celestial globes. Thus we meet with a star marked 84 
H., meaning that this is its number in Herschel's catalogue, or 
140 M. denoting the place the star occupies in the catalogue of 
Mayer. 

406. The earliest catalogue of the stars was made by Hippar- 
chus of the Alexandrian School, about 140 years before the 
Christian era. A new star appearing in the firmament, he was 
induced to count the stars and to record their positions, in order 
that posterity might be able to judge of the permanency of the con- 

32 



250 FIXED STARS. 

stellations. His catalogue contains all that were conspicuous to 
the naked eye in the latitude of Alexandria, being 1022. Most per- 
sons unacquainted with the actual number of the stars which com- 
pose the visible firmament, would suppose it to be much greater than 
this ; but it is found that the catalogue of Hipparchus embraces 
nearly all that can now be seen in the same latitude, and that on 
the equator, when the spectator has the northern and southern 
hemispheres both in view, the number of stars that can be counted 
does not exceed 3000. A careless view of the firmament in a 
clear night, gives us the impression of an infinite multitude of stars ; 
but when we begin to count them, they appear much more 
sparsely distributed than we supposed, and large portions of the 
sky appear almost destitute of stars. 

By the aid of the telescope, new fields of stars present them- 
selves of boundless extent ; the number continually augmenting 
as the powers of the telescope are increased. Lalande, in his 
Histoire Celeste, has registered the positions of no less than 
50,000 ; and the whole number visible in the largest telescopes 
amount to many millions. 

407. It is strongly recommended to the learner to acquaint 
himself with the leading constellations at least, and with a few 
of the most remarkable individual stars. The task of learning 
them is comparatively easy, and hardly any kind of knowledge, 
attained with so little labor, so amply rewards the possessor. It 
will generally be advisable, at the outset, to get some one already 
acquainted with the stars, to point out a few of the most conspicu- 
ous constellations, those of the Zodiac for example : the learner 
may then resort to a celestial globe,* and fill up the outline by 
tracing out the principal stars in each constellation as there laid 
down. By adding one new constellation to his list every night, 
and reviewing those already acquired, he will soon become fa- 
miliar with the stars, and will greatly augment his interest and 
improve his intelligence in celestial observation and practical as- 
tronomy. 

- * For the method of rectifying the globe so as to represent the appearance of the 
heavens on any particular evening, see page 27, Prob. 76. 



CONSTELLATIONS. 251 



CONSTELLATIONS. 



408. We will point out particular marks by which the leading 
constellations may be recognized, leaving it to the learner, after 
he has found a constellation, to trace out additional members of 
it by the aid of the celestial globe, or by maps of the stars. Let 
us begin with the Constellations of the Zodiac, which succeeding 
each other as they do in a known order, are most easily found. 

Aries (The Ram) is a small constellation, known by two bright 
stars which form his head, a and fi Arietis. These two stars are 
four degrees* apart ; and directly south of {3 at the distance of one 
degree, is a smaller star, y Arietis. It has been already inti- 
mated (Art. 193,) that the vernal equinox probably was near the 
head of Aries, when the signs of the Zodiac received the present 
names. 

Taurus (The Bull) will be readily found by the seven stars or 
Pleiades, which lie in his neck. The largest star in Taurus is 
Aldebaran, in the Bull's eye, a star of the first magnitude, of a 
reddish color somewhat resembling the planet Mars. Aldebaran 
and four other stars in the face of Taurus, compose the Hyades. 

Gemini (The Twins) is known by two very bright stars, Castor 
and Pollux, five degrees asunder. Castor (the northern) is of the 
first, and Pollux of the second magnitude. 

Cancer (The Crab). There are no large stars in this constel- 
lation, and it is regarded as less remarkable than any other in the 
Zodiac. It contains however an interesting group of small stars, 
called Prccsepe or the Nebula of Cancer, which resembles a comet, 
and is often mistaken for one, by persons unacquainted with 
the stars. With a telescope of very moderate powers this nebula 
is converted into a beautiful assemblage of exceedingly bright stars. 

Leo (The Lion) is a very large constellation, and has many 
interesting members. Regulus (a Leonis) is a star of the first 
magnitude, which lies directly in the ecliptic, and is much used in 
astronomical observations. North of Regulus lies a semi-circle of 
bright stars, forming a sickle of which Regulus is the handle. 



* These measures are not intended to be stated with exactness, but only with such 
a degree of accuracy as may serve for a general guide. 



252 FIXED STARS. 

Denebola, a star of the second magnitude, is in the Lion's tail, 25° 
northeast of Regulus. 

Virgo (The Virgin) extends a considerable way from west 
to east, but contains only a few bright stars. Spica, however, is 
a star of the first magnitude, and lies a little east of the place of 
the autumnal equinox. Eighteen degrees eastward of Denebola, 
and twenty degrees north of Spica, is Vindemiatrix, in the arm of 
Virgo, a star of the third magnitude. 

Libra (The Balance) is distinguished by three large stars, of 
which the two brightest constitute the beam of the balance, and 
the smallest forms the top or handle. 

Scorpio (The Scorpion) is one of the finest of the constella- 
tions. His head is formed of five bright stars arranged in the 
arc of a circle, which is crossed in the center by the ecliptic nearly 
at right angles, near the brightest of the five, (3 Scorpionis. Nine 
degrees southeast of this, is a remarkable star of the first magni- 
tude, of a reddish color, called Cor Scorpionis. or Antares. South 
of this a succession of bright stars sweep round towards the east, 
terminating in several small stars, forming the tail of the Scorpion. 

Sagittarius (The Archer). Northeast of the tail of the Scor- 
pion, are three stars in the arc of a circle which constitute the bow 
of the Archer, the central star being the brightest, directly west 
of which is a bright star which forms the arrow. 

Capricornus (The Goat) lies northeast of Sagittarius, and is 
known by two bright stars, three degrees apart, which form the 
head. 

Aquarius (The Water Bearer) is recognized by two stars 
in a line with a Capricorni, forming the shoulders of the figure. 
These two stars are 10° apart, and 3° southeast is a third star, 
which together with the other two, makes an acute triangle, of 
which the westernmost is the vertex. 

Pisces (The Fishes) lie between Aquarius and Aries. They 
are not distinguished by any large stars, but are connected by a 
series of small stars, that form a crooked line between them. 
Piscis Australia, the Southern Fish, lies directly below Aquarius, 
and is known by a single bright star far in the south, having a 
declination of 30°. The name of this star is Fomalhaut, and is 
much used in astronomical measurements. 



CONSTELLATIONS. 253 

409. The Constellations of the Zodiac, being first well learned, 
so as to be readily recognized, will facilitate the learning of others 
that lie north and south of them. Let us therefore next review 
the principal Northern Constellations, beginning north of Aries and 
proceeding from west to east. 

Andromeda, is characterized by three stars of the second mag- 
nitude, situated in a straight line, extending from west to east. 
The middle star is about 17° north of (3 Arietis. It is in the girdle 
of Andromeda, and is named Mirach. The other two lie at about 
equal distances, 14° west and east of Mirach. The western star, 
in the head of Andromeda, lies in the Equinoctial Colure. The 
eastern star, Almaak, is situated in the foot. 

Perseus lies directly north of the Pleiades, and contains sev- 
eral bright stars. About 18° from the Pleiades is Algol, a star 
of the second magnitude, in the Head of Medusa, which forms a 
part of the figure ; and 9° northeast of Algol is Algenib, of the 
same magnitude in the back of Perseus. Between Algenib and 
the Pleiades are three bright stars, at nearly equal intervals, which 
compose the right leg of Perseus. 

Auriga (the Wagoner) lies directly east of Perseus, and extends 
nearly parallel to that constellation from north to south. Capella, 
a very white and beautiful star of the first magnitude, distinguishes 
this constellation. The feet of Auriga are near the Bull's Horns. 

The Lynx comes next, but presents nothing particularly inter- 
esting, containing no stars above the fourth magnitude. 

Leo Minor consists of a collection of small stars north of the 
sickle in Leo, and south of the Great Bear. Its largest star is 
only of the third magnitude. 

Coma Berenices is a cluster of small stars, north of Denebola, 
in the tail of the lion, and of the head of Virgo. About 12 
directly north of Berenice's Hair, is a single bright star called Cor 
Caroli, or Charles's Heart. 

Bootes, which comes next, is easily found by means of Arc- 
turns, a star of the first magnitude, of a reddish color, which is 
situated near the knee of the figure. Arcturus is accompanied 
by three small stars forming a triangle a little to the southwest. 
Two bright stars y and 8 Bootis, form the shoulders, and /3 of the 
third magnitude is in the head of the figure. 



254 FIXED STARS. 

Corona Borealis (The Crown) which is situated E. of Bootes, 
is very easily recognized, composed as it is of a semi-circle of 
bright stars. In the center of the bright crown, is a star of the 
second magnitude, called gemma ; the remaining stars are all much 
smaller. 

Hercules, lying between the Crown on the west and the Lyre 
on the east, is very thickly set with stars, most of which are quite 
small. This Constellation covers a great extent of the sky, es- 
pecially from N. to S., the head terminating within 15° of the 
equator, and marked by a star of the third magnitude, called Ras- 
algethi, which is the largest in the Constellation. 

Ophiuchus is situated directly south of Hercules, extending some 
distance on both sides of the equator, the feet resting on the Scor- 
pion. The head terminates near the head of Hercules, and like 
that, is marked by a bright star within 5° of a Herculis. Ophiu- 
chus is represented as holding in his hands the Serpent, the head 
of which, consisting of three bright stars, is situated a little south 
of the Crown. The folds of the serpent will be easily followed 
by a succession of bright stars which extend a great way to the 
east. 

Aquila (The Eagle) is conspicuous for three bright stars in its 
neck, of which the central one, Altair, is a very brilliant white 
star of the first magnitude. Antinous lies directly south of the 
Eagle, and north of the head of Capricornus. 

Delphinus (The Dolphin) is a small but beautiful Constellation, 
a few degrees east of the Eagle, and is characterized by four bright 
stars near to one another, forming a small rhombic square. An- 
other star of the same magnitude 5° south, makes the tail. 

Pegasus lies between Aquarius on the southwest and Andromeda 
on the northeast. It contains but few large stars. A very regu- 
lar square of bright stars is composed of a. Andromedce, and the 
three largest stars in Pegasus, namely, Scheat, Markab, and Alge- 
nib. The sides composing this square are each about 15°. Alge- 
nib is situated in the equinoctial colure. 

410. We may now review the Constellations which surround 

the North Pole, within the circle of perpetual apparition. (Art. 54.) 

Ursa Minor (The Little Bear) lies nearest the pole. The 



CONSTELLATIONS. 255 

Pole-star, Polaris, is in the extremity of the tail, and is of the third 
magnitude. Three stars in a straight line 4° or 5° apart, com- 
mencing with the Pole-star, lead to a trapezium of four stars, and 
the whole seven form together a dipper, the trapezium being the 
body, and the three stars the handle. 

Ursa Major (The Great Bear) is situated between the pole 
and the Lesser Lion, and is usually recognized by the figure of a 
larger and more perfect dipper, which constitutes the hinder part 
of the animal. This has also seven stars, four in the body of the 
dipper, and three in the handle. All these are stars of much ce- 
lebrity. The two in the western side of the dipper, a and (3, are 
called Pointers, on account of their always being in a right line 
with the Pole-star, and therefore affording an easy mode of finding 
that. The first star in the tail, next the body, is named Alioth, and 
the second Mizar. The head of the Great Bear lies far to the 
westward of the Pointers, and is composed of numerous small 
stars ; and the feet are severally composed of two small stars very 
near to each other. 

Draco (The Dragon) winds round between the Great and Lit- 
tle Bear ; and commencing with the tail, between the Pointers and 
the Pole-star, it is easily traced by a succession of bright stars ex- 
tending from west to east ; passing under Ursa Minor, it returns 
westward, and terminates in a triangle which forms the head of 
Draco, near tho feet of Hercules, northwest of Lyra. 

Cepheus lies eastward of the breast of the Dragon, but has no 
stars above the third magnitude. 

Cassiopeia is known by the figure of a chair, composed of four 
stars which form the legs, and two which form the back. This 
Constellation lies between Perseus and Cepheus, in the Milky 
Way. 

Cygnus (The Swan) is situated also in the Milky Way, some 
distance southwest of Cassiopeia, towards the Eagle. Three 
bright stars, which lie along the Milky Way, form the body and 
neck of the Swan, and two others in a line with the middle one of 
the three, one above and one below, constitute the wings. This 
Constellation is among the few that exhibit some resemblance to 
the animals whose names they bear. 

Lyra (The Lyre) is directly west of the Swan, and is easily 



256 FIXED STARS. 

distinguished by a beautiful white star of the first magnitude, a 
Lyrce. 

411. The Southern Constellations are comparatively few in 
number. We shall notice only the Whale, Orion, the Greater and 
Lesser Dog, Hydra, and the Crow. 

Cetus (The Whale) is distinguished rather for its extent thanks 
brilliancy, reaching as it. does through 40° of longitude, while none 
of its stars except one, are above the third magnitude. Menkar (a 
Ceti) in the mouth, is a star of the second magnitude, and several 
other bright stars directly south of Aries, make the head and neck 
of the Whale. Mir a (o Ceti) in the neck of the Whale, is a varia 
ble star. 

Orion is one of the largest and most beautiful of the constella- 
tions, lying southeast of Taurus. A cluster of small stars form the 
head ; two large stars, Betalgeus of the first and Bellatrix of the 
second magnitude, make the shoulders ; three more bright stars 
compose the buckler, and three the sword ; and Rigel, another 
star of the first magnitude, makes one of the feet. In this Con- 
stellation there are 70 stars plainly visible to the naked eye, inclu- 
ding two of the first magnitude, four of the second, and three of 
the third. 

Canis Major lies S. E. of Orion, and is distinguished chiefly by 
its containing the largest of the fixed stars, Sirius. 

Canis Minor, a little north of the equator, between Canis Major 
and Gemini, is a small Constellation, consisting chiefly of two 
stars, of which Procyon is of the first magnitude. 

Hydra has its head near Procyon, consisting of a number of 
stars of ordinary brightness. About 15° S. E. of the head, is a 
star of the second magnitude, forming the heart, (Cor Hydrce) ; 
and eastward of this, is a long succession of stars of the fourth and 
fifth magnitudes composing the body and the tail, and reaching a 
few degrees south of Spica Virginis. 

Corvus (The Crow) is represented as standing on the tail of 
Hydra. It consists of small stars, only three of which are as 
large as the third magnitude. 

412. The foregoing brief sketch is designed merely to aid the 



CLUSTERS OF STARS. 257 

student in finding the principal constellations and the largest fixed 
stars. When we have once learned to recognize a constellation 
by some characteristic marks, we may afterwards fill up the out- 
line by the aid of a celestial globe or a map of the stars. It will be 
of little avail, however, merely to commit this sketch to memory ; 
but it will be very useful for the student at once to render himself 
familiar with it, from the actual specimens which every clear 
evening presents to his view. 



CHAPTER II. 

CLUSTERS OF STARS NEBULAE VARIABLE STARS TEMPORARY 

STARS DOUBLE STARS. 

413. In various parts of the firmament are seen large groups or 
clusters, which, either by the naked eye, or by the aid of the small- 
est telescope, are perceived to consist of a great number of small 
stars. Such are the Pleiades, Coma Berenices, and Praesepe or 
the Bee-hive, in Cancer. The Pleiades, or Seven Stars, as they 
are called, in the neck of Taurus, is the most conspicuous cluster. 
When we look directly at this group, we cannot distinguish 
more than six stars, but by turning the eye sideways* upon it, we 
discover that there are many more. Telescopes show 50 or 60 
stars crowded together and apparently insulated from the other 
parts of the heavens, f Coma Berenices has fewer stars, but they 
are of a larger class than those which compose the Pleiades. The 
Bee-hive or Nebula of Cancer as it is called, is one of the finest 
objects of this kind for a small telescope, being by its aid con- 
verted into a rich congeries of shining points. The head of Orion 
affords an example of another cluster, though less remarkable than 
the others. 

* Indirect vision is far more delicate than direct. Thus we can see the Zodiacal 
Light or a Comet's Tail, much more distinctly and better defined, if we fix one eye on 
a part of the heavens at some distance, and turn the other eye obliquely upon the ob- 
ject, t Sir J. Herschel. 
• 33 



258 FIXED STARS. 

414. Nebulce are those faint misty appearances which resemble 
comets, or a small speck of fog. The Galaxy or Milky Way pre- 
sents a continued succession of large nebulae. A very remarkable 
Nebula, visible to the naked eye, is seen in the girdle of Androme- 
da. No powers of the telescope have been able to resolve this 
into separate stars. Its dimensions are astonishingly great. In 
diameter it is about 15'. The telescope reveals to us innumerable 
objects of this kind. Sir William Herschel has given catalogues 
of 2000 Nebulae, and has shown that the nebulous matter is dis- 
tributed through the immensity of space in quantities inconceiva- 
bly great, and in separate parcels of all shapes and sizes, and of 
all degrees of brightness between a mere milky appearance and 
the condensed light of a fixed star. Finding that the gradations 
between the two extremes were tolerably regular, he thought it 
probable that the nebulae form the materials out of which nature 
elaborates suns and systems ; and he conceived that, in virtue of 
a central gravitation, each parcel of nebulous matter becomes 
more and more condensed, and assumes a rounder form. He in- 
fers from the eccentricity of its shape, and the effects of the mutual 
gravitation of its particles, that it requires gradually a rotary mo- 
tion ; that the condensation goes on increasing until the mass 
acquires consistency and solidity, and all the other characters of a 
comet or a planet ; that by a still further process of condensation, 
the body becomes a real star, self-shining ; and that thus the waste 
of the celestial bodies, by the perpetual diffusion of their light, is 
continually compensated and restored by new formations of such 
bodies, to replenish forever the universe with planets and stars.* 

415. These opinions are recited here rather out of respect to 
their notoriety and celebrity, than because we suppose them to be 
founded on any better evidence than conjecture. The Philosophi- 
cal Transactions for many years, both before and after the com- 
mencement of the present century, abound with both the ob- 
servations and speculations of Sir William Herschel. The for- 
mer are deserving of all praise ; the latter of less confidence. 
Changes, however, are going on in some of the nebulae, which 

* Phil. Trans. 1811. 



NEBULJE. 



259 



plainly show that they are not, like planets and stars, fixed and 
permanent creations. Thus the great nebula in the girdle of An- 
dromeda, has very much altered its structure since it first became 
an object of telescopic observation.* Many of the nebulae are of 
a globular form, (Fig. 72, a) but frequently they present the ap- 
(Fig. 72, a) (Fig.72, b.) 




pearance of a rapid increase of numbers towards the center, (Fig. 
72, b) the exterior boundary being irregular, and the central parts 
more nearly spherical. 

416. The Nebula in the sword of Orion is particularly celebrated, 
being very large and of a peculiarly interesting appearance. Ac- 
cording to Sir John Herschcl, its nebulous character is very dif- 
ferent from what might be supposed to arise from the assemblage 
of an immense collection of small stars. It is formed of little 
flocculent masses like wisps of clouds ; and such wisps seem to 
adhere to many small stars at its outskirts, and especially to one 
considerable star which it envelops with a nebulous atmosphere of 
considerable extent and singular figure. 

Descriptions, however, can convey but a very imperfect idea 
of this wonderful class of astronomical objects, and we would 
therefore urge the learner studiously to avail himself of the first 
opportunity he may have to view them through a large telescope, 
especially the Nebula of Andromeda and of Orion. 

417. Nebulous Stains are such as exhibit a sharp and brilliant 
star surrounded by a disk or atmosphere of nebulous matter. 
These atmospheres in some cases present a circular, in others an 
oval figure ; and in some instances, the nebula consists of a long, 
narrow spindle-shaped ray, tapering away at both ends to points. 



* Astron. Trans. II, 495. 



260 FIXED STARS. 

Annular Nebulce also exist, but are among the rarest objects in 
the heavens. The most conspicuous of this class, is to be found 
exactly half way between the stars (3 and y Lyrae, and may be seen 
with a telescope of moderate power.* 

Planetary Nebulce constitute another variety, and are very re- 
markable objects. They have, as their name imports, exactly the 
appearance of planets. Whatever may be their nature, they must 
be of enormous magnitude. One of them is to be found in the 
parallel of v Aquarii, and about 5m. preceding that star. Its appa- 
rent diameter is about 20". Another in the Constellation An- 
dromeda, presents a visible disk of 12", perfectly defined and 
round. Granting these objects to be equally distant from us with 
the stars, their real dimensions must be such as, on the lowest 
computation, would fill the orbit of Uranus. It is no less evident 
that, if they be solid bodies, of a solar nature, the intrinsic splendor 
of their surfaces must be almost infinitely inferior to that of the 
sun. A circular portion of the sun's disk, subtending an angle of 
20", would give a light equal to 100 full moons; while the objects 
in question are hardly, if at all, discernible with the naked eye.f 

418. The Galaxy or Milky Way is itself supposed by some to 
be a nebula of which the sun forms a component part ; and hence 
it appears so much greater than other nebulae only in consequence 
of our situation with respect to it, and its greater proximity to our 
system. So crowded are the stars in some parts of this zone, that 
Sir William Herschel, by counting the stars in a single field of his 
telescope, estimated that 50,000 had passed under his review in a 
zone two degrees in breadth during a single hour's observation. 
Notwithstanding the apparent contiguity of the stars which crowd 
the galaxy, it is certain that their mutual distances must be incon- 
ceivably great. 

419. Variable Stars are those which undergo a periodical 



* A list of 288 bright nebulae, with references to well known stars, near which they 
are situated, is given in the Edinburg Enyclopaedia, Art. Astronomy \ p. 781. It is con- 
venient for finding any required nebula. 

t Sir J. Herschel. 



TEMPORARY STARS. 261 

change of brightness. One of the most remarkable is the star 
Mira in the Whale, (o Ceti.) It appears once in 11 months, re- 
mains at its greatest brightness about a fortnight, being then, on 
some occasions, equal to a star of the second magnitude. It then 
decreases about three months, until it becomes completely invisible, 
and remains so about five months, when it again becomes visible, 
and continues increasing during the remaining three months of its 
period. 

Another very remarkable variable star is Algol ((3 Persei.) It 
is usually visible as a star of the second magnitude, and continues 
such for 2d. 14h. when it suddenly begins to diminish in splendor, 
and in about Si hours is reduced to the fourth magnitude. It then 
begins again to increase, and in 3| hours more, is restored to its 
usual brightness, going through all its changes in less than three 
days. This remarkable law of variation appears strongly to sug- 
gest the revolution round it of some opake body, which, when in- 
terposed between us and Algol, cuts off a large portion of its light. 
It is (says Sir J. Herschel) an indication of a high degree of ac- 
tivity in regions where, but for such evidences, we might conclude 
all lifeless. Our sun requires almost nine times this period to per- 
form a revolution on its axis. On the other hand, the periodic 
time of an opake revolving body, sufficiently large, which would 
produce a similar temporary obscuration of the sun, seen from a 
fixed star, would be less than fourteen hours. 

The duration of these periods is extremely various. While that 
of (3 Persei above mentioned, is less than three days, others are 
more than a year, and others many years. 

420. Temporary Stars are new stars which have appeared 
suddenly in the firmament, and after a certain interval, as suddenly 
disappeared and returned no more. 

It was the appearance of a new star of this kind 125 years be- 
fore the Christian era, that prompted Hipparchus to draw up a 
catalogue of the stars, the first on record. Such also was the star 
which suddenly shone out A. D. 389, in the Eagle, as bright as 
Venus, and after remaining three weeks, disappeared entirely. 
At other periods, at distant intervals, similar phenomena have pre- 
sented themselves. Thus the appearance of a star in 1572, was 



262 FIXED STARS. 

so sudden, that Tycho Brahe returning home one day was sur- 
prized to find a collection of country people gazing at a star, which 
he was sure did not exist half an hour before. It was then as 
bright as Sirius, and continued to increase until it surpassed Jupi- 
ter when brightest, and was visible at mid-day. In a month it 
began to diminish, and in three months afterwards it had entirely 
disappeared. 

It has been supposed by some that in a few instances, the same 
star has returned, constituting one of the periodical or variable 
stars of a long period. 

Moreover, on a careful re-examination of the heavens, and a 
comparison of catalogues, many stars are now found to be miss- 
ing.* 

421. Double Stars are those which appear single to the naked 
eye, but are resolved into two by the telescope ; or if not visible 
to the naked eye, are seen in the telescope very close together. 
Sometimes three or more stars are found in this near connexion, 
constituting triple or multiple stars. Castor, for example, when 
seen by the naked eye, appears as a single star, but in a telescope 
even of moderate powers, it is resolved into two stars of between 
the third and fourth magnitudes, within 5" of each other. These 
two stars are nearly of equal size, but frequently one is exceedingly 
small in comparison with the other, resembling a satellite near its 
primary, although in distance, in light, and in other characteristics, 
each has all the attributes of a star, and the combination therefore 
cannot be that of a planet with a satellite. In some instances, also, 
the distance between these objects is much less than 5", and in 
many cases it is less than 1". The extreme closeness, together 
with the exceeding minuteness of most of the double stars, requires 
the best telescope united with the most acute powers of observa- 
tion. Indeed, certain of these objects are regarded as the severest 
tests both of the excellence of the instrument, and of the skill of the 
observer. 

422. Many of the double stars exhibit the curious and beautiful 

* Sir J. Herschel. 



DOUBLE STARS. 263 

phenomena of contrasted or complementary colors* In such in- 
stances, the larger star is usually of a ruddy or orange hue, while 
the smaller one appears blue or green, probably in virtue of that 
general law of optics, which provides that when the retina is ex- 
cited by any bright colored light, feebler lights which seen alone 
would produce no sensation but of whiteness, appear colored, with 
the tint complementary to that of the brighter. Thus a yellow 
color predominating in the light of the brighter star, that of the 
less bright one in the same field of view will appear blue ; while, 
if the tint of the brighter star verges to crimson, that of the other 
will exhibit a tendency to green, or even under favorable circum- 
stances, will appear as a vivid green. The former contrast is 
beautifully exhibited by » Cancri, the latter by 7 Andromeda?, both 
fine double stars. If, however, the colored star is much the 
less bright of the two, it will not materially affect the other. Thus 
for instance, r\ Cassiopeia? exhibits the beautiful combination of a 
large white star, and a small one of a rich ruddy purple. It is by 
no means, however, intended to say, that in all such cases, one of 
the colors is the mere effect of contrast, and it may be easier sug- 
gested in words, than conceived in imagination, what variety of 
illumination two suns, a red and green, or a yellow and a blue sun, 
must afford a planet circulating about either ; and what charming 
contrasts and " grateful vicissitudes," a red and green day for in- 
stance, alternating with a white one and with darkness, might arise 
from the presence or absence of one or other, or both above the 
horizon. Insulated stars of a red color, almost as deep as that of 
blood, occur in many parts of the heavens, but no green or blue star, 
of any decided hue, has, we believe, ever been noticed, unassociated 
with a companion brighter than itself, f 

423. Our knowledge of the double stars almost commenced with 
Sir William Herschel, about the year 1780. At the time he 



* Complementary colors are such as together make white light. If all the colors of 
the spectrum be laid down on a circular ring, each occupying its proportionate space, any 
two colors on the opposite sides of the zone, are complementary to each other, and 
when of the same degree of intensity, they compose white light. Brewster's Optics, 
Part 111, c. 26. 

t Sir J. Herschel. 



264 FIXED STARS. 

began his search for them, he was acquainted with only fcrur. 
Within five years, he discovered nearly 700 double stars.* In his 
memoirs published in the Philosophical Transactions,! he gave 
most accurate measurements of the distances between the two 
stars, and of the angle which a line joining the two, formed with 
the parallel of declination.J These data would enable him, or at 
least posterity, to judge whether these minute bodies ever change 
their position with respect to each other. 

Since 1821, these researches have been prosecuted with great 
zeal and industry by Sir James South and Sir John Herschel in 
England, and by Professor Struve at Dorpat in Russia, and the 
whole number of double stars now known, amounts to several 
thousands. § Two circumstances add a high degree of interest 
to the phenomena of the double stars, — the first is, that a few of 
them at least are found to have a revolution around each other, and 
the second, that they are supposed to afford the means of obtain- 
ing the parallax of the fixed stars. Of these topics we shall treat 
in the next chapter. 



CHAPTER III. 

MOTIONS OF THE FIXED STARS DISTANCES NATURE. 

424. In 1803, Sir William Herschel first determined and an- 
nounced to the world, that there exist among the stars, separate 
systems, composed of two stars revolving about each other in regu- 
lar orbits. These he denominated Binary Stars, to distinguish 
them from other double stars where no such motion is detected, 
and whose proximity to each other may possibly arise from casual 
juxta-position, or from one being in the range of the other. Be- 

* During his life he observed in all, 2400 double stars. 

t Phil. Trans. 1782—1785. X Baily, Astron. Trans, u. 542. 

§ The Catologue of Struve, contains 3063. 



MOTIONS OF THE FIXED STARS. 



265 



tween fifty and sixty instances of changes to a greater or less 
amount of the relative position of double stars, are mentioned by 
Sir William Herschel ; and a few of them had changed their places 
so much within 25 years, and in such order, as to lead him to the 
conclusion that they perform revolutions, one around the other, 
in regular orbits. 

425. These conclusions have been fully confirmed by later ob- 
servers, so that it is now considered as fully established, that there 
exist among the fixed stars, binary systems, in which two stars 
perform to each other the office of sun and planet, and that the 
periods of revolution of more than one such pair have been ascer- 
tained with something approaching to exactness. Immersions and 
emersions of stars behind each other have been observed, and real 
motions among them detected rapid enough to become sensible 
and measurable in very short intervals of time.* The following 
table exhibits the present state of our knowledge on this subject. 



Names. 


Period in years. 


Major axis of the orbit. 


Eccentricity. 


y) Coronae, 

£ Cancri, 

I Ursae Majoris, 


43.40 
55.00 
58.26 










7".714 


0.4164 


70 Ophiuchi, 


80.34 


8.784 


0.4667 


Castor, 


252.66 


16.172 


0.7582 


<f Coronae, 


286.00 


7.358 


0.6112 


61 Cygni, 


452.00 


30.860 






y Virginis, 


628.90 


24.000 


0.8335 


7 Leonis, 


1200.00 











From this table it appears (1) that the periods of the double stars 
are very various, ranging in the case of those already ascertained 
from forty three years to one thousand ; (2) that their orbits are 
very small ellipses more eccentric than those of the planets, the 
greatest of which (that of Mercury) having an eccentricity of only 
about .2 of the major axis. 

The most remarkable of the binary stars is y Virginis, on account 
not only of the length of its period, but also of the great diminution 
of apparent distance, and rapid increase of angular motion about 
each other of the individuals composing it. It is a bright star 



* Ast. Trans, ii, 544. 
34 



266 FIXED STARS. 

of the fourth magnitude, and its component stars are almost ex- 
actly equal. It has been known to consist of two stars since the 
beginning of the eighteenth century, their distance being then 
between six and seven seconds ; so that any tolerably good tele- 
scope would resolve it. Since that time, they have been con- 
stantly approaching, and were in 1838 hardly more than a single 
second asunder ; so that no telescope that is not of a very supe- 
rior quality, is competent to show them otherwise than as a single 
star, somewhat lengthened in one direction. It fortunately hap- 
pens that Bradley (Astronomer Royal) in 1718, noticed, and re- 
corded in the margin of one of his observation books, the apparent 
direction of their line of junction, as being parallel to that of two 
remarkable stars a and 5 of the same constellation, as seen by the 
naked eye, — a remark which has been of signal service in the 
investigation of their orbit. It is found that it passed its perihelion 
in June, 1836. The period given in the table is that assigned 
by Sir John Herschel ; but later observations indicate a much 
shorter period. In the interval from 1839 to 1841, the star | Ursae 
Majoris, completed a full revolution from the epoch of the first 
measurement of its position in 1781 ; and the regularity with which 
it has maintained its motion, is said to have been exceedingly 
beautiful.* 

426. The revolutions of the binary stars have assured us of 
that most interesting fact, that the law of gravitation extends to 
the fixed stars. Before these discoveries, we could not decide 
except by a feeble analogy that this law transcended the bounds 
of the solar system. Indeed, our belief of the fact rested more 
upon our idea of unity of design in all the works of the Creator, 
than upon any certain proof; but the revolution of one star around 
another in obedience to forces which must be similar to those that 
govern the solar system, establishes the grand conclusion, that the 
law of gravitation is truly the law of the material universe. 

We have the same evidence (says Sir John Herschel) of the 
revolutions of the binary stars about each other, that we have of 
those of Saturn and Uranus about the sun ; and the correspond- 

* Ast. Trans, v, 35. 



MOTIONS OF THE FIXED STARS. 267 

ence between their calculated and observed places in such elon- 
gated ellipses, must be admitted to carry with it a proof of the 
prevalence of the Newtonian law of gravity in their systems, of 
the very same nature and cogency as that of the calculated and 
observed places of comets round the center of our own system. 

But (he adds) it is not with the revolutions of bodies of a plan- 
etary or cometary nature round a solar center that we are now 
concerned ; it is with that of sun around sun, eaoh, perhaps, ac- 
companied with its train of planets and their satellites, closely 
shrouded from our view by the splendor of their respective suns, 
and crowded into a space, bearing hardly a greater proportion to 
the enormous interval which separates them, than the distances of 
the satellites of our planets from their primaries, bear to their dis- 
tances from the sun itself. 

427. Some of the fixed stars appear to have a real motion in space. 

The apparent change of place in the stars arising from the pre- 
cession of the equinoxes, the nutation of the earth's axis, the dimi- 
nution of the obliquity of the ecliptic, and the aberration of light, 
have been already mentioned ; but after all these corrections are 
made, changes of place still occur, which cannot result from any 
changes in the earth, but must arise from changes in the stars 
themselves. Such motions are called the proper motions of the 
stars. Nearly 2000 years ago, Hipparchus and Ptolemy made the 
most accurate determinations in their power of the relative situa- 
tions of the stars, and their observations have been transmitted to us 
in Ptolemy's Almagest ; from which it appears that the stars retain 
at least very nearly the same places now as they did at that period. 
Still, the more accurate methods of modern astronomers, have 
brought to light minute changes in the places of certain stars 
which force upon us the conclusion, either that our solar system 
causes an apparent displacement of certain stars, by a motion of its 
own in space, or that they have themselves a proper motion. Pos- 
sibly, indeed, both these causes may operate. 

428. If the sun, and of course the earth which accompanies 
him, is actually in motion, the fact may become manifest from 
the apparent approach of the stars in the region which he is leav- 



268 FIXED STARS. 

ing, and the recession of those which lie in the part of the heav- 
ens towards which he is travelling. Were two groves of trees 
situated on a plain at some distance apart, and we should go from 
one to the other, the trees before us would gradually appear fur- 
ther and further asunder, while those we left behind would appear 
to approach each other. Some years since, Sir William Herschel 
supposed he had detected changes of this kind among two sets of 
stars in opposite points of the heavens, and announced that the 
solar system was in motion towards a point in the constellation 
Hercules ;* but other astronomers have not found the changes 
in question such as would correspond to this motion, or to any 
motion of the sun ; and while it is a matter of general belief that 
the sun has a motion in space, the fact is not considered as yet 
entirely proved. 

429. In most cases where a proper motion in certain stars has 
been suspected, its annual amount has been so small, that many 
years are required to assure us, that the effect is not owing to 
some other cause than a real progressive motion in the stars them- 
selves ; but in a few instances the fact is too obvious to admit of 
any doubt. Thus the two stars 61 Cygni, which are nearly equal, 
have remained constantly at the same, or nearly at the same dis- 
tance of 15" for at least fifty years past. Meanwhile they have 
shifted their local situation in the heavens, 4' 23", the annual 
proper motion of each star being 5."3, by which quantity this 
system is every year carried along in some unknown path, by a 
motion which for many centuries must be regarded as uniform 
and rectilinear. A greater proportion of the double stars than 
of any other indicate proper motions, especially the binary stars 
or those which have a revolution around each other. Among 
stars not double, and no way differing from the rest in any other 
obvious particular, ^ Cassiopeiae has the greatest proper motion of 
any yet ascertained, amounting to nearly 4" annually. 

DISTANCES OF THE FIXED STARS. 

430. We cannot ascertain the actual distance of any of the fixed 

* Phil. Trans. 1783, 1805, and 1806. 



DISTANCES OF THE FIXED STARS. 269 

stars, but can certainly determine that the nearest star is more than 
(20,000,000,000,000,) twenty billions of miles from the earth. 

For all measurements relating to the distances of the sun and 
planets, the radius of the earth furnishes the base line (Art. 87.) 
The length of this line being known, and the horizontal parallax 
of the body whose distance is sought, we readily obtain the dis- 
tance by the solution of a right angled triangle. But any star 
viewed from the opposite sides of the earth, would appear from 
both stations, to occupy precisely the same situation in the celes- 
tial sphere, and of course it would exhibit no horizontal parallax. 

But astronomers have endeavored to find a parallax in some of 
the fixed stars by taking the diameter of the earth 's orbit as a base 
line. Yet even a change of position amounting to 190 millions 
of miles, proves insufficient to alter the place of a single star, from 
which it is concluded that the stars have not even any annual par- 
allax ; that is, the angle subtended by the semi-diameter of the 
earth's orbit, at the nearest fixed star, is insensible. The errors to 
which instrumental measurements are subject, arising from the 
defects of the instruments themselves, from refraction, and from 
various other sources of inaccuracy, are such, that the angular 
determinations of arcs of the heavens cannot be relied on to less 
than 1". But the change of place in any star when viewed at 
opposite extremities of the earth's orbit, is less than 1", and there- 
fore cannot be appreciated by direct measurement. It follows, 
that, when viewed from the nearest star, the diameter of the earth's 
orbit would be insensible ; the spider line of the telescope would 
more than cover it. 

431. Taking, however, the annual parallax of a fixed star at 1", 
let a b (Fig. 73) represent the radius of the earth's orbit and c a 
fixed star, the angle at c being 1" and the angle at b a right angle ; 
then, 

Sin. 1" : Rad.::l : 200,000, nearly. 

Hence the hypothenuse of a triangle whose vertical angle is 
1" is about 200,000 times the base; consequently the distance of 
the nearest fixed star must exceed 95000000 x200000=190000000 x 
100000, or one hundred thousand times one hundred and ninety 
millions of miles. Of a distance so vast we can form no adequate 



270 FIXED STARS. 

conceptions, and even seek to measure it only by the time 
that light, (which moves more than 192,000 miles per lg ' c 
second and passes from the sun to the earth in 8m. 13.3 
sec.,) would take to traverse it, which is found to be more 
than three and a half years. 

If these conclusions are drawn with respect to the 
largest of the fixed stars, which we suppose to be vastly 
nearer to us than those of the smallest magnitude, the idea 
of distance swells upon us when we attempt to estimate 
the remoteness of the latter. As it is uncertain, however, 
whether the difference in the apparent magnitudes of the 
stars is owing to a real difference or merely to their being 
at various distances from the eye, more or less uncertainty 
must attend all efforts to determine the relative distances 
of the stars ; but astronomers generally believe that the 
lower orders of stars are vastly more distant from us than the 
higher. Of some stars it is said, that thousands of years would 
be required for their light to travel down to us. 

432. We have said that the stars have no annual parallax ; yet 
it may be observed that astronomers are not exactly agreed on 
this point. Dr. Brinkley, a late eminent Irish astronomer, sup- 
posed that he had detected an annual parallax in a Lyrae amounting 
to l".l 3 and in a Aquilse of 1"A2.* These results were contro- 
verted by Mr. Pond of the Royal Observatory of Greenwich ; and 
Mr. Struve of Dorpat has shown that in a number of cases, the 
parallax is negative, that is, in a direction opposite to that which 
would arise from the motion of the earth. Hence it is considered 
doubtful whether in all cases of an apparent parallax, the effect is 
not wholly due to errors of observation 

433. Indirect methods have been proposed for ascertaining the 
parallax of the fixed stars by means of observations on the double 
stars. If the two stars composing a double star are at different 
distances from us, parallax would affect them unequally, and change 
their relative position with respect to each other ; and since the 

* Phil. Trans. 1821. 



NATURE OF THE FIXED STARS. 271 

ordinary sources of error arising from the imperfection of instru- 
ments, from precession, nutation, aberration, and refraction, would 
be avoided, (as they would affect both objects alike, and there- 
fore would not disturb their relative positions,) measurements 
taken with the micrometer of changes much less than 1" may be 
relied on. Sir John Herschel proposes a method* by which 
changes may be determined which amount to only T V of a second.f 

434. The immense distance of the fixed stars is inferred also 
from the fact that the largest telescopes do not increase their ap- 
parent magnitude. They are still points, when viewed with the 
highest magnifiers, although they sometimes present a spurious 
disk, which is owing to irradiation. J 

NATURE OF THE STARS. 

435. The stars are bodies greater than our earth. If this were 
not the case they could not be visible at such an immense distance. 
Dr. Wollaston, a distinguished English philosopher, attempted to esti- 
mate the magnitudes of certain of the fixed stars from the light which 
they afford. By means of an accurate photometer (an instrument 
for measuring the relative intensities of light) he compared the 
light of Sirius with that of the sun. He next inquired how far the 
sun must be removed from us in order to appear no brighter than 
Sirius. He found the distance to be 141,400 times its present dis- 
tance. But Sirius is more than 200,000 times as far off as the sun 
(Art. 431.) Hence he inferred that, upon the lowest computation, 
Sirius must actually give out twice as much light as the sun ; or 
that, in point of splendor, Sirius must be at least equal to two 

* Phil. Trans. 1826. 

t Very recent intelligence informs us, that Professor Bessel of Konigsberg, has ob- 
tained decisive evidence of an annual parallax in 61 Cygni, amounting to 0."3136. 
This makes the distance of that star, equal to 657700 times 95 millions of miles, — a 
distance which it would take light 10.3 years to traverse. 

X Irradiation is an enlargement of objects beyond their proper bounds, in consequence 
of the vivid impression of light on the eye. It is supposed to increase the apparent di- 
ameters of the sun and moon from three to four seconds, and to create an appearance 
of a disk in a fixed star which, when this cause is removed, is seen as a mere point. See 
Richardson, Astr. Trans, v, 1. 



272 FIXED STARS. 

suns. Indeed, he has rendered it probable that the light of Sirius 
is equal to fourteen suns. 

436. The fixed stars are suns. We have already seen that they 
are large bodies; that they are immensely further off than the 
furthest planet; that they shine by their own light: in short, that 
their appearance is, in all respects, the same as the sun would ex- 
hibit if removed to the region of the stars. Hence we infer that 
they are bodies of the same kind with the sun. 

437. We are justified therefore by a sound analogy, in concluding 
that the stars were made for the same end as the sun, namely, as 
the centers of attraction to other planetary worlds, to which they 
severally dispense light and heat. Although the starry heavens 
present, in a clear night, a spectacle of ineffable grandeur and 
beauty, yet it must be admitted that the chief purpose of the stars 
could not have been to adorn the night, since by far the greatest 
part of them are wholly invisible to the naked eye ; nor as land- 
marks to the navigator, for only a very small proportion of them 
are adapted to this purpose ; nor, finally, to influence the earth by 
their attractions, since their distance renders such an effect entirely 
insensible. If they are suns, and if they exert no important agen- 
cies upon our world, but are bodies evidently adapted to the same 
purpose as our sun, then it is as rational to suppose that they were 
made to give light and heat, as that the eye was made for seeing 
and the ear for hearing. It is obvious to inquire next, to what 
they dispense these gifts if not to planetary worlds ; and why to 
planetary worlds, if not for the use of percipient beings ? We 
are thus led, almost inevitably, to the idea of a Plurality of 
Worlds ; and the conclusion is forced upon us, that the spot which 
the Creator has assigned to us is but a humble province of his 
boundless empire.* 

* See this argument, in its full extent, in Dick's Celestial Scenery. 



CHAPTER IV. 



OF THE SYSTEM OF THE WORLD. 



438. The arrangement of all the bodies that compose the material 
universe, and their relations to each other, constitute the System of 
the World. 

It is otherwise called the Mechanism of the Heavens ; and in- 
deed in the System of the world, we figure to ourselves a machine, 
all the parts of which have a mutual dependence, and conspire to 
one great end. " The machines that are first invented (says Adam 
Smith) to perform any particular movement, are always the most 
complex ; and succeeding artists generally discover that with fewer 
wheels and with fewer principles of motion than had originally 
been employed, the same effects may be more easily produced. 
The first systems, in the same manner, are always the most com- 
plex ; and a particular connecting chain or principle is generally 
thought necessary to unite every two seemingly disjointed appear- 
ances ; but it often happens, that one great connecting principle is 
afterwards found to be sufficient, to bind together all the discord- 
ant phenomena that occur in a whole species of things." This re- 
mark is strikingly applicable to the origin and progress of systems 
of astronomy. 

439. From the visionary notions which are generally understood 
to have been entertained on this subject by the ancients, we are 
apt to imagine that they knew less than they actually did of the 
truths of astronomy. But Pythagoras, who lived 500 years before 
the Christian era, was acquainted with many important facts in 
our science, and entertained many opinions respecting the system 
of the world which are now held to be true. Among other things 
well known to Pythagoras were the following : 

1. The principal Constellations. These had begun to be formed 
in the earliest ages of the world. Several of them bearing the 
same names as at present are mentioned in the writings of Hesiod 

35 



274 system or THE world. 

and Homer ; and the " sweet influences of the Pleiades" and the 
" bands of Orion," are beautifully alluded to in the book of Job. 

2. Eclipses. Pythagoras knew both the causes of eclipses and 
how to predict them ;* not indeed in the accurate manner now 
employed, but by means of the Saros (Art. 233.) 

3. Pythagoras had divined the true system of the world, hold- 
ing that the sun and not the earth, (as was generally held by the 
ancients, even for many ages after Pythagoras,) is the center 
around which all the planets revolve, and that the stars are so many 
suns, each the center of a system like our own.f Among lesser 
things, he knew that the earth is round ; that its surface is naturally 
divided into five Zones ; and that the ecliptic is inclined to the 
equator. He also held that the earth revolves daily on its axis, and 
yearly around the sun ; that the galaxy is an assemblage of small 
stars ; and that it is the same luminary, namely, Venus, that con- 
stitutes both the morning and the evening star, whereas all the an- 
cients before him had supposed that each was a separate planet, 
and accordingly the morning star was called Lucifer, and the 
evening star Hesperus.J He held also that the planets were in- 
habited, and even went so far as to calculate the size of some of 
the animals in the moon.§ Pythagoras was so great an enthusiast 
in music, that he not only assigned to it a conspicuous place in his 
system of education, but even supposed the heavenly bodies them- 
selves to be arranged at distances corresponding to the diatonic 
scale, and imagined them to pursue their sublime march to notes 
created by their own harmonious movements, called the " music 
of the spheres;" but he maintained that this celestial concert, 
though loud and grand, is not audible to the feeble organs of man, 
but only to the gods. 

440. With few exceptions, however, the opinions of Pythago- 
ras on the System of the World, were founded in truth. Yet they 
were rejected by Aristotle and by most succeeding astronomers 
down to the time of Copernicus, and in their place was substituted 



* Long's Astronomy, 2. 671. 

t Library of Useful Knowledge, History of Astronomy. 

X Long's Ast. 2. 673. § Ed. Encyclopaedia. 



ASTRONOMICAL KNOWLEDGE OF THE ANCIENTS. 275 

the doctrine of Cyrstalline Spheres, first taught by Eudoxus. Ac- 
cording to this system, the heavenly bodies are set like gems in 
hollow solid orbs, composed of crystal so pellucid that no anterior 
orb obstructs in the least the view of any of the orbs that lie behind 
it. The sun and the planets have each its separate orb ; but the 
fixed stars are all set in the same grand orb ; and beyond this is 
another still, the Primum Mobile, which revolves daily from east 
to west, and carries along with it all the other orbs. Above the 
whole, spreads the Grand Empyrean, or third heavens, the abode 
of perpetual serenity.* 

To account for the planetary motions, it was supposed that each 
of the planetary orbs as well as that of the sun, has a motion of its 
own eastward, while it partakes of the common diurnal motion of 
the starry sphere. Aristotle taught that these motions are effected 
by a tutelary genius of each planet, residing in it, and directing its 
motions, as the mind of man directs his motions. 

441. On coming down to the time of Hipparchus, who flourished 
about 150 years before the Christian era, we meet with rstrono- 
mers who acquired far more accurate knowledge of the celestial 
motions. Hipparchus was in possession of instruments for meas- 
uring angles, and knew how to resolve spherical triangles. He 
ascertained the length of the year within 6m. of the truth. He 
discovered the eccentricity of the solar orbit, (although he supposed 
the sun actually to move uniformly in a circle, but the earth to be 
placed out of the center,) and the positions of the sun's apogee and 
perigee. He formed very accurate estimates of the obliquity of 
the ecliptic and of the precession of the equinoxes. He computed 
the exact period of the synodic revolution of the moon, and the 
inclination of the lunar orbit ; discovered the motion of her node 
and of her line of apsides ; and made the first attempts to ascer- 
tain the horizontal parallaxes of the sun and moon. 

Such was the state of astronomical knowledge when Ptolemy 
wrote the Almagest, in which he has transmitted to us an ency- 
clopaedia of the astronomy of the ancients. 



* Long's Ast. 2. 640— Robinson's Mech. Phil. 2. 83— Gregory's Ast. 132— Playfair's 
Dissertation, 118. 



276 SYSTEM OF THE WORLH. 

442. The systems of the world which have been most celebrated 
are three — the Ptolemaic, the Tychonic, and the Copernican. 
We shall conclude this part of our work with a concise statement 
and discussion of each of these systems of the Mechanism of the 
Heavens. 

THE PTOLEMAIC SYSTEM. 

443. The doctrines of the Ptolemaic System were not originated 
by Ptolemy, but being digested by him out of materials furnished 
by various hands, it has come down to us under the sanction of 
his name. 

According to this system, the earth is the center of the universe, 
and all the heavenly bodies daily revolve around it from east to 
west. In order to explain the planetary motions, Ptolemy had re- 
course to deferents and epicycles, — an explanation devised by Apol- 
lonius, one of the greatest geometers of antiquity.* He conceived 
that, in the circumference of a circle, having the earth for its cen- 
ter, there moves the center of another circle, in the circumference 
of which the planet actually revolves. The circle surrounding the 
earth was called the deferent, while the smaller circle, whose center 
was always in the periphery of the deferent, was called the epi- 
cycle. The motion in each was supposed to be uniform. Lastly, 
it was conceived that the motion of the center of the epicycle in 
the circumference of the deferent, and of the deferent itself, are 
in opposite directions, the first being towards the east, and the 
second towards the west. 

444. But these views will be better understood from a diagram. 
Therefore, let ABC (Fig. 74,) represent the deferent, E being the 
earth a little out of the center. Let abc represent the epicycle, 
having its center at v, on the periphery of the deferent. Con- 
ceive the circumference of the deferent to be carried about the 
earth every twenty-four hours in the order of the letters ; and at 
the same time, let the center v of the epicycle abed, have a slow 
motion in the opposite direction, and let a body revolve in this cir- 

* Playfair, Dissertation Second, 119. 



THE PTOLEMAIC SYSTEM. 277 

cle in the direction abed. Then it will be seen that the body 
would actually describe the looped curves klmnop ; that it would 

(Fig. 74.) 




appear stationary at I and m, and at n and o; that its motion 
would be direct from k to I, and then retrograde from I to m ; di- 
rect again from m to n, and retrograde from n to o. 

445. Such a deferent and epicycle may be devised for each 
planet as will fully explain all its ordinary motions ; but it is in- 
consistent with the phases of Mercury and Venus, which being be- 
tween us and the sun on both sides of the epicycle, would present 
their dark sides towards us in both these positions, whereas at one 
of the conjunctions they are seen to shine with full face.* It is 
moreover absurd to speak of a geometrical center, which has no 
bodily existence, moving around the earth on the circumference 
of another circle ; and hence some suppose that the ancients 
merely assumed this hypothesis as affording a convenient geome- 
trical representation of the phenomena, — a diagram simply, with- 
out conceiving the system to have any real existence in nature. 

* Vince's Complete System, I, 96. 



278 SYSTEM OF THE WORLD. 

446. The objections to the Ptolemaic system, in general, are the 
following : First, it is a mere hypothesis, having no evidence in its 
favor, except that it explains the phenomena. This evidence is 
insufficient of itself, since it frequently happens that each of two 
hypotheses, directly opposite to each other, will explain all the 
known phenomena. But the Ptolemaic system does not even do 
this, as it is inconsistent with the phases of Mercury and Venus, 
as already observed. Secondly, now that we are acquainted with 
the distances of the remoter planets, and especially of the fixed 
stars, the swiftness of motion implied in a daily revolution of the 
starry firmament around the earth, renders such a motion wholly 
incredible. Thirdly, the centrifugal force that would be generated 
in these bodies, especially in the sun, renders it impossible that 
they can continue to revolve around the earth as a center. 

These reasons are sufficient to show the absurdities of the 
Ptolemaic System of the World. 



THE TYCHONIC SYSTEM. 

447. Tycho Brahe, like Ptolemy, placed the earth in the center 
of the universe, and accounted for the diurnal motions in the same 
manner as Ptolemy had done, namely, by an actual revolution of 
the whole host of heaven around the earth every twenty-four 
hours. But he rejected the scheme of deferents and epicycles, and 
held that the moon revolves about the earth as the center of her 
motions ; that the sun, and not the earth, is the center of the plan- 
etary motions ; and that the sun accompanied by the planets 
moves around the earth once a year, somewhat in the manner that 
we now conceive of Jupiter and his satellites as revolving around 
the sun. 

448. The system of Tycho serves to explain all the common 
phenomena of the planetary motions, but it is encumbered with 
the same objections as those that have been mentioned as resting 
against the Ptolemaic system, namely, that it is a mere hypothesis ; 
that it implies an incredible swiftness in the diurnal motions ; and 
that it is inconsistent with the known laws of universal gravitation. 



THE COPERNICAN SYSTEM. 279 

But if the heavens do not revolve, the earth must, and this brings 
us to the system of Copernicus. 

THE COPERNICAN SYSTEM. 

449. Copernicus was born at Thorn in Prussia in 1473. The 
system that bears his name was the fruit of forty years of intense 
study and meditation upon the celestial motions. As already men- 
tioned, (Art. 6,) it maintains (1) That the apparent diurnal motions 
of the heavenly bodies, from east to west, is owing to the real 
revolution of the earth on its own axis from west to east ; and (2) 
That the sun is the center around which the earth and planets all 
revolve from west to east. It rests on the following arguments : 

450. First, the earth revolves on its own axis. 

1. Because this supposition is vastly more simple. 

2. It is agreeable to analogy, since all the other planets that 
afford any means of determining the question, are seen to revolve 
on their axes. 

3. The spheroidal figure of the earth, is the figure of equilibrium, 
that results from a revolution on its axis. 

4. The diminished weight of bodies at the equator, indicates a 
centrifugal force arising from such a revolution. 

5. Bodies let fall from a high eminence, fall eastward of their 
base, indicating that when further from the center of the earth 
they were subject to a greater velocity, which in consequence of 
their inertia, they do not entirely lose in descending to the lower 
level.* 

451. Secondly, the planets, including the earth, revolve about the 
sun. 

1. The phases of Mercury and Venus are precisely such, as would 
result from their circulating around the sun in orbits within that 
of the earth ; but they are never seen in opposition, as they would 
be if they circulated around the earth. 

2. The superior planets do indeed revolve around the earth ; 
but they also revolve around the sun, as is evident from their phases 

* Biot. 



280 SYSTEM OF THE WORLD. 

and from the known dimensions of their orbits ; and that the sun 
and not the earth, is the center of their motions, is inferred from 
the greater symmetry of their motions as referred to the sun than 
as referred to the earth, and especially from the laws of gravitation, 
which forbid our supposing that bodies so much larger than the 
earth, as some of these bodies are, can circulate permanently around 
the earth, the latter remaining all the while at rest. 

3. The annual motion of the earth itself is indicated also by the 
most conclusive arguments. For, first, since all the planets with 
their satellites, and the comets, revolve about the sun, analogy 
leads us to infer the same respecting the earth and its satellite. 
Secondly, the motions of the satellites, as those of Jupiter and 
Saturn, indicate that it is a law of the solar system that the smaller 
bodies revolve about the larger. Thirdly, on the supposition that 
the earth performs an annual revolution around the sun, it is em- 
braced along with the planets, in Kepler's law, that the squares of 
the times are as the cubes of the distances ; otherwise, it forms an 
exception, and the only known exception to this law. Lastly, the 
aberration of light affords a sensible proof of the motion of the 
earth, since that phenomenon indicates both a progressive motion 
of light, and a motion of the earth from west to east. (Art. 195.) 

452. It only remains to inquire, whether there subsist higher 
orders of relations between the stars themselves. 

The revolutions of the binary stars (Art. 424) afford conclusive 
evidence of at least subordinate systems of suns, governed by the 
the same laws as those which regulate the motions of the solar 
system. The nebulce also compose peculiar systems, in which the 
members are evidently bound together by some common relation. 

In these marks of organization, — of stars associated together in 
clusters, — of sun revolving around sun, and of nebulae disposed 
in regular figures, we recognize different members of some grand 
system, links in one great chain that binds together all parts of 
the universe ; as we see Jupiter and his satellites combined in one 
subordinate system, and Saturn and his satellites in another, — each 
a vast kingdom, and both uniting with a number of other indi- 
vidual parts to compose an empire still more vast. 



THE COPERNICAN SYSTEM. 281 

453. This fact being now established, that the stars are immense 
bodies like the sun, and that they are subject to the laws of gravi- 
tation, we cannot conceive how they can be preserved from falling 
into final disorder and ruin, unless they move in harmonious con- 
cert like the members of the solar system. Otherwise, those that 
are situated on the confines of creation, being retained by no forces 
from without, while they are subject to the attraction of all the 
bodies within, must leave their stations, and move inward with 
accelerated velocity, and thus all the bodies in the universe would 
at length fall together in the common center of gravity. The 
immense distance at which the stars are placed from each other, 
would indeed delay such a catastrophe ; but such must be the ulti- 
mate tendency of the material world, unless sustained in one har- 
monious system by nicely adjusted motions.* To leave entirely 
out of view our confidence in the wisdom and preserving goodness 
of the Creator, and reasoning merely from what we know of the 
stability of the solar system, we should be justified in inferring, 
that other worlds are not subject to forces which operate only to 
hasten their decay, and to involve them in final ruin. 

We conclude, therefore, that the material universe is one great 
system ; that the combination of planets with their satellites con- 
stitutes the first or lowest order of worlds ; that next to these 
planets are linked to suns ; that these are bound to other suns, 
composing a still higher order in the scale of being ; and, finally, 
that all the different systems of worlds, move around their common 
center of gravity. 

* Robinson's Physical Astronomy. 
36 



Note on Meteoric Showers. — (See p. 77.) 

The remarkable exhibitions of shooting stars which have occur- 
red within a few years past, have excited great interest among 
astronomers, and led to some new views respecting the construc- 
tion of the solar system. Their attention was first turned towards 
this subject by the great meteoric shower of November 13th, 
1833. On that morning, from two o'clock until broad day light, 
the sky being perfectly serene and cloudless, the whole heavens 
were lighted with a magnificent display of celestial fireworks. 
At times, the air was filled with streaks of light, occasioned by 
fiery particles darting down so swiftly as to leave the impression 
of their light on the eye, (like a match ignited and whirled before 
the face,) and drifting to the northwest like flakes of snow driven 
by the wind ; while, at short intervals, balls of fire varying in 
size from minute points to bodies larger than Jupiter and Venus, 
and in a few instances as large as the full moon, descended more 
slowly along the arch of the sky, often leaving after them long 
trains of light, which were, in some instances, variegated with dif- 
ferent prismatic colors. 

On tracing back the lines of direction in which the meteors 
moved, it was found that they all appeared to radiate from the 
same point, which was situated near one of the stars (y Leonis) 
of the sickle, in the constellation Leo ; and, in every repetition of 
the meteoric shower of November, the radiant point has occupied 
nearly the same situation. 

This shower pervaded nearly the whole of North America, hav- . 
ing appeared in almost equal splendor from the British possessions 
on the north, to the West India islands and Mexico on the south, 
and from sixty-one degrees of longitude east of the American 
coast, quite to the Pacific ocean on the west. Throughout this im- 
mense region, the duration was nearly the same. The meteors 
began to attract attention by their unusual frequency and bril- 
liancy, from nine to twelve o'clock in the evening ; were most 
striking in their appearance from two to four ; arrived at their 
maximum, in many places, about four o'clock ; and continued 
until rendered invisible by the light of day. The meteors moved 
in right lines, or in such apparent curves, as, upon optical princi- 
ples, can be resolved into right lines. Their general tendency 
was towards the northwest, although by the effect of perspective 
they appeared to move in all directions. 

Soon after this occurrence, it was ascertained that a similar 
meteoric shower had appeared in 1799, and what was remarkable, 



NOTE ON METEORIC SHOWERS. 283 

almost exactly at the same time of year, namely, on the morning 
of the 12th of November ; and it soon appeared, by accounts 
received from different parts of the world, that this phenomenon 
had occurred on the same 13th of November, in 1830, 1831, and 
1832. Hence, this w T as evidently an event independent of the 
casual changes of the atmosphere ; for, having a periodical return, 
it was undoubtedly to be referred to astronomical causes, and its 
recurrence, at a certain definite period of the year, plainly indi- 
cated some relation to the revolution of the earth around the sun. 

It remained, however, to develop the nature of this relation, by 
investigating, if possible, the origin of the meteors. The views to 
which the author of this work was led, suggested the probability 
that the same phenomenon would recur at the corresponding sea- 
sons of the year, for at least several years afterwards ; and such 
proved to be the fact, although the appearances, at every succeed- 
ing return, were less and less striking, until 1839, when, so far as 
is known, they ceased altogether. 

Meanwhile, three other distinct periods of meteoric showers have 
also been determined ; one about the 21st of April, another on the 
9th of August, and another on the 7th of December. 

The following conclusions respecting the meteoric shower of 
November, are believed to be well established, and most of them 
are now generally admitted by astronomers, though we cannot here 
exhibit the evidence on which they were founded, but must beg 
leave to refer the reader to various publications on this subject in 
the American Journal of Science, commencing with the twenty-fifth 
volume ; and also to " Letters on Astronomy," by the author of 
this work. 

It is considered, then, as established, that the meteors had their 
origin beyond the limits of the atmosphere, having descended to 
us from some body existing in space independent of the earth ; 
that they consisted of exceedingly light combustible matter ; that 
they moved with very great velocities, amounting in some in- 
stances to not less than fourteen miles per second ; that some of 
them were bodies of large size, probably several hundred feet in 
diameter; that when they entered the atmosphere, they rapidly 
and powerfully condensed the air before them, and thus elicited 
the heat which set them on fire, as a spark is sometimes evolved 
by condensing air suddenly by a piston and cylinder ; and that 
they were consumed and resolved into small clouds at the height 
of about thirty miles above the earth. 

Calling the body from which the meteors descended the " me- 
teoric body," it is inferred that it is a body of great extent, since, 
without apparent exhaustion, it has been able to afford such copious 
showers of meteors at so many different times ; and hence we 
regard the part that has descended to the earth only as the ex- 
treme portions of a body or collection of meteors, of unknown 



284 NOTE ON METEORIC SHOWERS. 

extent, existing in the planetary spaces. Since the earth fell in 
with the meteoric body, in the same part of its orbit for several 
years in succession, the body must either have remained there 
while the earth was performing its whole revolution around the 
sun, or it must itself have had a revolution, as well as the earth. 
No body can remain stationary within the planetary spaces ; for, 
unless attracted to some nearer body, it would be drawn directly 
towards the sun, and could not have been encountered by the 
earth again in the same part of her orbit. Nor can any mode 
be conceived in which this event could have happened so many 
times in regular succession, unless the body had a revolution of 
its own around the sun. Finally, to have come into contact with 
the earth at the same part of her orbit, in two or more suc- 
cessive years, the body must have either a period which is nearly 
the -same with the earth's period, or some aliquot part of it. No 
period will fulfil the conditions, but either a year or half a year. 
Which of these is the true period of the meteoric body, is not 
fully determined. 



PLATE I. 
1. Castor. 2.7Leonis. 3. 39Drac. 4. X Oph. 5. llMonoc. 6.£Cancri. 




7. 4 & 5 s Lyrae, et Debilissima. 



8. tf Orionis. 




9. y Virginis. 



71 

1837. 1838. 1839. 1840. 1845. 1850. 1860. Orbit. 

Fig. 10. 



g BS B 






JU; 


if: 


j 


c 


I 


e 


1 


l 


1 


I 


J 


'II* L 





INTRODUCTION 



PRACTICAL ASTRONOMY, 



DESIGNED AS A 



SUPPLEMENT TO OLMSTED'S ASTRONOMY; 



CONTAINING SPECIAL RULES FOR THE 



ADJUSTMENT AND USE OF ASTRONOMICAL INSTRUMENTS, 



TOGETHER WITH THE 



CALCULATION OF ECLIPSES AND OCCULTATIONS, 



AND THE METHODS OF FINDING THE 



LATITUDE AND LONGITUDE 



BY EBENEZER PORTER MASON. 



NEW YORK: COLLINS, KEESE, & Co, 



TRINTED BY B. L. HAMLEN — NEW HAVEN, CONN. 



1841. 



Entered according to Act of Congress, in the year 1841, by 

B. L. Hamlen, 
in the Clerk's office, of the District Court of Connecticut. 



ADVERTISEMENT. 



The lamented author of the following Treatise, the late 
Ebenezer Porter Mason, in early youth developed talents for 
Practical Astronomy so extraordinary, as to leave no doubt, that 
had his life been spared, he would have risen to a rank among 
the first astronomers of the age. But he became a martyr to his 
zeal, and died of consumption on the 26th of December, 1840, 
in the twenty second year of his age. He had already acquired 
consummate skill in astronomical observations, and had made as- 
tonishing proficiency in the knowledge of the heavenly bodies. 

The most competent judges have pronounced this treatise to 
be unrivalled in its kind, supplying as it does, in a small com- 
pass, a great amount of useful information respecting the struc- 
ture, adjustment, and management, of astronomical instruments, 
and affording examples of the best methods of making astronom- 
ical calculations. The kind of information, moreover, is precise- 
ly such as is needed by the young astronomer. Those who have 
arrived at eminence in this difficult department of scientific labor, 
by a long career of persevering efforts, usually find themselves 
in a labyrinth, where they are unable to retrace their steps, and 
to place themselves again at the entrance of their course ; but 
young Mason, by an extraordinary flight of genius, rose so rap- 
idly to the highest walks of astronomy, that the various expedi- 
ents which he had employed to surmount the obstacles that suc- 
cessively stood in his path, were still fresh in his memory, and 
he was, therefore, able to take the young aspirant by the hand, 
and conduct him through the dark and intricate way which he 
himself had recently threaded with such wonderful success. 

The closing part of this article was the last fruit of his genius, 
the concluding paragraph having been written only three weeks 
before his death. This is no place for writing his eulogy ; but 



VI ADVERTISEMENT. 

science will long mourn the loss of a youth signally endowed 
with that rare assemblage of qualities, essential to the structure 
of the great astronomer, in which are united the refined artist 
and the profound mathematician. 

The author of the preceding " Introduction to Astronomy," is 
therefore happy to be able to follow the study of the elements of 
the science, which he has endeavored to expound to his pupils, 
with the practical application, in a form so well fitted to awaken 
their zeal, and reward their industry. 



ANALYSIS. 



Page. 

CHAPTER I.— Telescope, 1 

General description, 2 

Refracting Telescope, 2 

Reflecting do 2 

Finder, 3 

Herschel's 40 ft. reflector, 4 

Dorpat refractor, 4 

Holcombe's Telescopes, 5 

Smith's 14 ft. reflector, 5 

Directions for choosing a telescope,. . 6 

Power and light of telescopes, 7 

Magnifying power how ascertained,. . 7 
Eye-pieces, their various combinations, 7 
Different powers for different purposes, 8 

Light of different telescopes, 9 

Impediments to high magnifying pow- 
ers, 10 

Apparent discs of stars, 12 

Comparison of refractors and reflect- 
ors, 12 

Test objects of telescopic power and 

light, 14 

List of test objects for telescopes of 

different foci, 16 

Description of the stars figured in the 

frontispiece, 18 

How to find objects for examination, 20 
Points of compass in the heavens,.... 21 

How to examine difficult objects, 23 

To remedy the defects of a telescope, 24 
How researches suggested, 25 

CHAPTER II.— Minor Astronom- 
ical Instruments, 28 

Micrometer, 29 

Vernier, 31 

Reacting microscopes, 32 

Level, 32 

To level the axis of a transit instru- 
ment, 32 

Plumb Line, 33 

Artificial Horizon, 34 

Floating Collimator, 34 

CHAPTER III. — Sextant— Alti- 
tude and Azimuth Instrument — 

Equatorial, 34 

Sextant, 34 

Adjustments, 35 



Pasre. 

To find the index error, 36 

To measure the semidiameter, 37 

To take altitudes by reflection, 37 

To find the distance between two ob- 
jects, 38 

Altitude and Azimuth Instrument,... 39 

Description and use, 39 

Advantages, 39 

Equatorial, 40 

Description and use, 40 

Adjustments, 40 

To find a star, 40 

To find R. Ascensions and Declina- 
tions, 40 

Advantages and disadvantages, 41 

CHAPTER IV. — Transit Instru- 
ment, 42 

Location, 42 

Adjustments, 43 

Distinctness of vision and parallax,... 44 

Horizontality of the axis, 45 

Perpendicularity of wires, 46 

Collimation in azimuth, 46 

Collimation in altitude, 46 

Position in the meridian, 47 

By the pole star, 47 

By circumpolar stars, 48 

By a high and a low star, 48 

Meridian Mark, 49 

Clock, 50 

Rate and Error, 51 

Regulation, 51 

Method of observing and registering 

transits, 52 

Equatorial interval, 53 

Reduction of observations to the mean 

wire, 55 

Correction of observations, 55 

Errors of the transit instrument and 

their correction, 58 

Formula? expressing the same, 61 

An evening's observations and their 
reduction, 65 

CHAPTER V.— Problems in Prac- 
tical Astronomy, 69 

Use of signs in connection with loga- 
rithms 69 



Vlll 



ANALYSIS. 



Page. 

Conversion of solar into sidereal time, 
and the contrary, 70 

Interpolation of differences, 72 

To find a maximum or minimum 
value, 74 

Corrections of the places of the fix- 
ed stars, 75 

CHAPTER VI.— Eclipses of the 

Moon, 77 

To find the places of the sun and 

moon, 77 

Comparison of the use of Nautical 

Almanac and Lunar Tables, 78 

Demonstration of formulae, 79 

Calculation of a lunar eclipse, 82 

To project a lunar eclipse, 85 

CHAPTER VII. — Occultations 

and Eclipses of the Sun, 86 

Occultations defined, 87 

Parallax of the moon, 88 

Principles of calculation, 88 

Parallaxes of the moon in right as- 

cension and declination, 89 

Reduction to the sphere, 90 

Augmentation of the moon's semi- 

d'ameter, 91 

Investigation of formulae for occul- 
tations, 92 

Investigation of formulae for a solar 

ecli pse, 96 

Synopsis of formulae for stellar and 

solar occultations, 102 

Calculation of a solar eclipse, 103 

Of the times of first and last contact, 103 
Of the time of greatest obscuration, 
and the minimum distance, 110 



Page. 

Formulae for the times of beginning 
and end in a total eclipse, and of 
formation and rupture of the ring 
in an annular eclipse, Ill 

Calculation of angles from" north point 
and vertex, 112 

To project an occultation or solar 
eclipse, 113 

Practical uses of calculating occul- 
tations and solar eclipses, 114 

Approximate calculation for observa- 
tion, 115 

CHAPTER VIII.— Methods of de- 
termining Latitudes and Longi- 
tudes, 118 

Methods of finding the Latitude,.... 118 
By altitudes of the pole star at any 

hour of the night, 119 

By meridian altitudes of the pole 

star, 119 

By meridian altitudes of the sun or 

any heavenly body, 119 

Examples for New Haven, 120 

By double alts, of the sun and moon, 121 
Methods of finding the Longitude,... 123 

By eclipses of the moon, 123 

do Jupiter's satellites, .... 123 

By signals, &c 124 

By chronometers, 124 

By occultations, 124 

Example, 126 

By Lunar distances, 128 

By moon-culminating stars, 129 

Example, 130 

Conclusion, 132 

Note, 134 

Tables, 137 



SUPPLEMENT 



OLMSTED'S ASTRONOMY. 



CHAPTER I. 



OF THE TELESCOPE. 



1. The application of the telescope to the purposes of prac- 
tical astronomy, is as varied as universal. When attached to a 
system of circles, and by connection with them compelled to 
trace out in its motion corresponding circles in the heavens, it 
appears under the various forms of a transit, equatorial, altitude 
and azimuth instrument, mural circle, and sextant. In such com- 
binations it is usually employed in exactly fixing the places of 
the heavenly bodies, and determining their distances from each 
other and from fixed points of reference. When unconnected 
with any such appendages, it becomes a simple telescope, free to 
sweep the heavens at large, and its principal use consists in the 
observation of phenomena and events which occur beyond our 
earth, and of the natural history of celestial objects. Under 
this, its simple form, and applied to observations of the latter 

kind, it will be considered in the present article. 

» 

2. We shall presume in the student an acquaintance with the 
principles of telescopic vision, and with the general construction 
of the different forms, under which the instrument has appeared 
since its first invention. That we may most simply and easily 
initiate him into the practice of astronomy, we shall suppose him 
at once an amateur observer, about to choose an instrument, and 
wishing to apply it in a way interesting to himself and useful to 
science. We shall direct our reader simply and plainly how to 

1 



2 THE TELESCOPE. 

choose his telescope, and judge of its excellence ; — how to remedy 
its defects, manage its adjustments, find objects which he wishes 
to examine, examine them to the best advantage,— and finally, 
what kind of observations to make, and in what way to proceed, 
if he would add utility to pleasure. 

3. General description of refracting and reflecting telescopes. 
The refracting telescope is usually mounted in a brass tube, 

and supported on a stand of various construction, — but oftenest 
a pillar of brass, with three folding supports. The object-glass 
must be achromatic, consisting of two (sometimes three) lenses 
so combined as to destroy very nearly the ill effects of color 
and aberration. The available diameter of the object-glass is 
called the aperture of the telescope, and is usually a little less 
than that of the tube. It forms its image near the eye-end of 
the telescope ; the distance between the object-glass and this 
image is called the focal length of the telescope, and is commonly 
rather greater than that of the main tube. A microscope, con- 
sisting of two or more small lenses, is applied at the eye-end to 
magnify this image as if it were a tangible object ; this is called 
an eye-piece, and with every large telescope, several of them are 
always furnished by the maker, to allow of variety in its magni- 
fying power. The eye-piece is set in a sliding tube, and thus 
pushed in and out from the main tube by a milled head, which 
controls a concealed rack and pinion. The object of this con- 
trivance is to enable the observer to adjust his microscope or 
eye-piece accurately upon the image, and is especially necessary 
where several of different foci are to be successively screwed 
into the eye tube. Such a telescope is pointed to any celestial 
object by wheeling it around the perpendicular support or axis 
till it arrives in the vertical plane of the star, and then turning it 
on a second pivot which allows of motion in altitude. If the 
instrument is of considerable magnitude and power, contrivances 
are attached, so that by a screw or pinion the telescope may have 
slow motions in altitude and azimuth, and yet may admit of dis- 
engagement when it is to be turned through any considerable arc. 

4. Reflecting telescopes are of four kinds ; the Newtonian, 
Herschelian, Gregorian and Cassegranian. Their principles of 



THE TELESCOPE. 3 

construction are usually explained in elementary works on natu- 
ral philosophy.* The Cassegranian is nearly obsolete ; it diners 
but slightly from the Gregorian, and both are mounted in much 
the same style as refracting telescopes. In the Newtonian form, 
however, the eye does not look in at the lower extremity, and a 
different kind of mounting is necessary. When pointed at a 
star, the pencil of rays reflected from the large mirror at the 
lower end of the tube is turned off at right angles by the inclined 
plane mirror, and an eye-piece on the side of the tube receives 
it. The small mirror and eye-piece are attached to a slide, 
which may be moved to and from the large mirror in adjusting 
the focus. 

In this form of the reflecting telescope, the eye looks at right 
angles to the tube of the' telescope. But in the Herschelian con- 
struction, the small mirror is removed, and the eye looks imme- 
diately down the tube towards the large mirror. In order, how- 
ever, that the head of the observer in this position may intercept 
no portion of the broad cylindrical beam of light, which enters 
the telescope from the star, the large mirror is slightly inclined 
so as to cast its focal image at the extreme left hand margin of 
the mouth of the instrument, where it is magnified by an eye- 
piece sliding along that side of the tube, and viewed with the 
right eye of the observer. It may be here remarked that it is a 
very common misconception to suppose that the interposition of 
the head or of any irregular body over the aperture of a tele- 
scope will obscure or cut out a corresponding portion of the disc 
of a star, of a planet or the sun ; but a little consideration of 
that part of optics which teaches, that each point of the image 
is formed by rays from every portion of the speculum, will show 
that the only effect of such interposition is to diminish propor- 
tionally the light of the image. 

5. Every telescope of considerable magnifying power should 
be furnished with a finder, which is a small telescope of a very 
low power, attached firmly to the side of the larger, and exactly 
parallel with it. In the common focus of the object and eye 
glass is a pair of coarse cross-wires. The necessity of this ap- 
pendage becomes evident when we reflect, that a high magnify- 

* Olmsted's Nat. Phil. Art. 918-920. 



4 THE TELESCOPE. 

ing power requires a very small actual field of view. Thus the 
field of a telescope magnifying between 100 and 200 times is a 
circle in the heavens not as large as the full moon, and will scarcely 
include the Pleiades or seven stars. The telescope is approxi- 
mately pointed to a star by glancing the eye along the tube ; the 
object then appears in the field of the finder, because its low 
magnifying power allows of a wide field of view of several de- 
grees in diameter. It may be easily brought upon the intersec- 
tion of the cross-wires in the finder, and will then be somewhere 
in the field of view of the telescope. If the smaller instrument 
is out of adjustment, however, it will not be the index of the 
larger, and perfect correspondence of direction between the two 
must be thus obtained ; — direct the telescope to a star, or better, 
to a distant terrestrial point not in motion, by the uncertain and 
tedious process of ranging for it till it is found ; then turn the 
screws which alter the position of the finder, or of its cross- 
wires, until the object is covered by the intersection of the latter. 
Ever after, the finder will be a sure guide to any other object. 

6. Description of particular refracting and reflecting telescopes. 

The largest reflecting telescope ever constructed was the cele- 
brated one of 40 ft. in focal length and 4 ft. aperture, by Sir 
Wo. Herschel. We shall occasionally allude to it, especially in 
speaking of the light of a telescope. It is now out of repair, in 
consequence of exposure to the weather ; and 20-ft. telescopes, 
of easier management, and more regular performance under 
changes of temperature and atmospheric equilibrium, are substi- 
tuted in its room. 

The difficulty of procuring discs of flint glass of sufficient size 
and homogeneity has prevented the construction of enormous re- 
fracting telescopes. A few, however, of nearly 1 foot in aper- 
ture, have been brought to high perfection at a great expense by 
the first artists of Europe, among which the celebrated Dorpat 
telescope, of 13 ft. focal length, and 10 inches aperture, has, until 
very lately, ranked the first. These large telescopes are mounted 
equatorially, and an apparatus of clock-work carries round the ho- 
rary circle and tube so as accurately to follow a star in its diurnal 
motion. Very recent experiments on the manufacture of flint glass 
promise the practicability of considerable increase in aperture. 



THE TELESCOPE. 5 

7. Such telescopes as these are, and will be for a time, to say 
the least, very rare in America. In confining our remarks more 
exclusively to those in use in this country, we shall mention par- 
ticularly the Herschelian telescopes of Mr. Holcombe, which 
are in very general use among observers, and from their excel- 
lence, will probably become much more so. For the benefit of 
the American observer, we will briefly describe its construction, 
and point out its advantages and disadvantages. It as of the 
Herschelian form, with a tube of sheet iron painted, and the end 
containing the large mirror rests on the ground. The other end 
is supported by two folding legs, which with the tube of the tel- 
escope form a kind of tripod. It is quickly directed to any 
point by spreading out more or less the two legs, and as each of 
these are double, and may be gradually lengthened or shortened 
at pleasure, a slow motion is obtained for following a star. The 
chief recommendation of this telescope is, that with an excel- 
lence of figure in the speculum which enables it to compete with 
the telescopes of the best European artists in its performance on 
the closest test-objects, it is afforded at a comparatively very mod- 
erate price, the cost of foreign instruments of equal power and 
light being two or three times greater. The telescope is per- 
fectly supported at precisely the two points most important, the 
place of the large mirror, and of the eye-piece ; its steadiness 
therefore gives it the advantage over most European instruments 
even actually superior, except in very calm nights. In elegance 
of external appearance, it will not compare with foreign instru- 
ments of equal intrinsic excellence ; the style of mounting, how- 
ever, is as neat and convenient as it is simple and inexpensive.* 
Other telescopes, American and foreign, but chiefly of the re- 
fracting kind, are frequently to be met with, whose astronomical 
excellence will recommend them to the notice of the observer. 



* The establishment of Mr. Holcombe is at Southwick, Mass., and his telescopes 
are from 5 to 14 feet in focal length, with apertures from 4 to 10 inches, at prices 
ranging from 100 to 600 dollars. The performance of some of these will be spoken 
of in our description of the test-objects of telescopes. 

The largest telescope in this country is a Herschelian of 14 feet focal length and 
12 inches aperture, principally constructed by Mr. Smith, lately a student of Yale Col- 
lege. It was first erected in New-Haven, Conn., but has recently been removed, and 
remounted at Ohio City, Ohio. 



O THE TELESCOPE. 

8. In the choice of an instrument, many particulars are to be 
attended to, among which we rank the following as the princi- 
pal. The telescope should show a star free from burs or tails 
of light, or haziness consequent on aberration. If an achro- 
matic, it should be noticed, whether the correction of color is 
perfect or not. In a good telescope, a slight alteration of no 
more than the T Vth of an inch in the focal adjustment, in 
either direction from the point of distinct vision, will consider- 
ably enlarge the image of a star, and render it indistinct. Day 
objects to an inexperienced observer, appear nearly as well in 
an ordinary instrument, as in an excellent one ; the moon and 
planets are much better objects of comparison, and a bright fixed 
star affords the severest trial of excellence, especially if it be 
double, like Castor or y Virginis. 

Such questions as these, " How far can I see with this tele- 
scope ?" or " How many times does it magnify ?" are not likely 
to elicit satisfactory information with regard to the optical capa- 
city of an instrument. For with an inferior telescope objects 
may be seen immeasurably remote, by directing it towards the 
stars ; and may be magnified almost any number of times, more 
or less distinctly, by applying eye-glasses of sufficient power. 
The highest power supplied with a telescope is commonly much 
greater than can be employed to the utmost advantage, and 
therefore is no very sure indication of its distinctness or defin- 
ing power. The true way of arriving at a knowledge of the 
excellence of an instrument is to find out what test-objects it has 
resolved, and is competent to exhibit clearly, either by inquiry, or 
actual observation. By referring to our table of test-objects in 
this chapter, and the remarks concerning them, the reader will 
have the best means of deciding on the merits of any telescope. 
The qualities of the stand should not be neglected, the essen- 
tial requisites of which are, — steadiness, ease of direction to any 
point of the heavens, and ease in following any celestial body in 
its diurnal motion. A star in the field of an unsteady telescope 
becomes like an irregularly whirled point of flame, upon which 
it is as impossible to exercise close observation, as to read from 
a printed leaf in rapid motion. Every tremor and vibration of 
the instrument is magnified by its whole optical power. A tel- 
escope supported at two points at a distance from each other, is 



THE TELESCOPE. 7 

far more steady than one balanced on its centre. The other two 
requisites are obtained when the telescope is capable of good 
quick and slow motions in directions at right angles to one an- 
other ; the quick motion affords the means of turning immedi- 
ately to the star, and the slower motion of following it with regu- 
larity. 

9. The power and light of telescopes. 

A telescope magnifies celestial objects in two ways. First, it 
expands their linear dimensions by what is called its magnifying 
power ; this power is expressed in numbers, and is the ratio of 
the visual angle under which the telescopic image is seen, to that 
under which the object appears to the naked eye. Secondly, it 
magnifies their light by what is termed its illuminating power ; 
which also may be expressed in numbers, and is nearly the ratio 
of a cylindrical beam of light, of the diameter of the object- 
glass or speculum, to the much narrower pencil of light, which 
is collected by the naked eye, and which is of the same diameter 
as the pupil. 

10. The magnifying power of a telescope may always be as- 
certained by dividing the solar focal length of the object-glass 
(S) by that of the eye-glass (<p). If the eye-piece is compound, 
and consists of two glasses, let F be the solar focal length of the 
inner,/ that of the outer glass, and d the distance between them ; 

Ff S 

then p — j. — -j=y, or the focal length of the equivalent lens ; and — 

will be the magnifying power, as before. The focus of any lens 
or mirror is found by casting the sun's image through or from it, 
on paper, moving the latter, till the image is most neatly defined 
and smallest. 

Every telescope is accompanied with a set of eye-pieces, fur- 
nishing an ascending series of powers. For day observation, a 
terrestrial eye-piece of low power, and consisting of four glasses 
set in a tube of considerable length, is useful to represent objects 
in an erect position. With celestial objects the inversion of the 
image is of little consequence, and short combinations of two 
glasses each are employed. These are called astronomical eye- 
pieces, and are of two kinds. In that invented by Huygens, the 



o THE TELESCOPE. 

focal length of the glass next the eye, the distance between the 
two, and the focal length of the other, are nearly as the numbers 
1, 2, and 3, and the plane surfaces of each lens are towards the 
eye. This eye-piece receives the pencil of rays from the object- 
glass before they reach the focus, and no aerial image is formed 
except between its two lenses ; it is therefore called a negative 
eye-piece, because, like a convex mirror, it has only a negative 
or imaginary focus. The positive eye-piece was invented by 
Ramsden, and consists of two lenses of nearly equal focus, and 
distant by f or f the focal length of either, with their convex 
surfaces towards each other. The image of the object-glass 
in this case is formed in the air just before the pencil of rays 
meets the inner glass of the eye-piece. Few other combinations 
of two glasses will afford good vision. Ramsden's eye-piece and 
the Huygenian are nearly equal in their merits. The latter af- 
fords perhaps a larger and more uniformly distinct field ; but the 
former admits of the insertion of micrometrical and other cross- 
lines at the place of the aerial image, and moreover, has what is 
called a flat field, — that is, does not distort the outline of an 
object. 

Very low powers, as from 30 to 80, are useful to view ill de- 
fined objects, as comets and many of the nebulae, — to command a 
large field of view, when desirable, — and to exhibit celestial ob- 
jects to those who are unaccustomed to the management of a 
telescope. As the magnifying power increases, the intrinsic 
brightness of the object is diminished. The planet Jupiter, for 
instance, sends an absolute quantity of light to the eye, which 
with the same aperture is constant under all powers. By mag- 
nifying his surface more and more, the light which belongs to it 
must necessarily be spread over that surface more and more 
thinly ; so that very soon a point is reached, where indistinctness 
begins, because the light at any one point of the retina is too 
faint to stimulate perfect vision. With a good 7-ft. achromatic 
or reflector, this cause alone renders 250 as high a power as can 
be employed to much advantage on the planets. An ordinary 
observer will be far better pleased by the view with 100 or 
120, because much brighter and apparently more distinct ; mi- 
nute features of the object may, however, be often better brought 
out by pushing the magnifying power even to indistinctness. 



THE TELESCOPE. 



Saturn will scarcely bear more than 200 well, because of its 
greater distance from the sun, and consequent feebleness of light ; 
the moon, on the contrary, will occasionally bear 300 in the in- 
spection of minute features, although 80 or 100 is always better 
to show it in whole. The fixed stars shine by their own intrinsic 
light, and will bear much higher powers. In very favorable cir- 
cumstances, and with an excellent telescope, some advantage may 
be gained with 600 or even 1000, at the expense of much aber- 
ration and troublesome indistinctness. Sir Wm. Herschel, with 
his exquisite 7-ft. Newtonian, occasionally on rare nights, pushed 
his powers even to 6000, but the common observer will do well if he 
ever gains any advantage by going beyond 600. For the planets, 
powers between 100 and 200 are most commonly available ; for 
double stars, those between 200 and 300, and if very close, 400 
or 500 may be employed. Single lenses are of advantage in 
very high powers, because there is less room for loss of light, 
for imperfections of density in the interior, and of figure in the 
surfaces, of one glass than of two. They are more difficult to 
use, since the object under inspection must be kept scrupulously 
in the middle of the field, both in adjustment of focus and in 
subsequent examination ; for in no other part of the field is there 
good vision. 

11. The light of a telescope depends chiefly on its aperture. 
Through a 7-ft. telescope with 150 Jupiter is seen with ease and 
comfort ; the same power on a 20-ft. telescope renders it so bright 
as to be painful to the eye, and too dazzling for distinctness. 
For this reason, the 7-ft. is preferable, unless the aperture of the 
20-ft. be sufficiently contracted. Again, the field of the 20-ft. is 
strown with stars, which are utterly invisible in the 7-ft ; here 
the 20-ft. has the advantage. Large telescopes and a great aper- 
ture are chiefly useful to view faint objects, as nebulse, and to 
bring out faint points of light. The 40-ft. of Sir Wm. Herschel 
would not bear as high a power as his 20-ft., and neither could 
at all admit of such high powers, as we have already said he some- 
times applied to his 7-ft. ; but yet the former, from its power 
of light alone, afforded him immediately several of his most cele- 
brated discoveries. 

When we consider that the pupil of the eye, and consequently 

2 



10 THE TELESCOPE. 

the pencil of rays which enters it from any star, is but 0.2 of 
an inch in diameter, we may conceive how greatly we should 
increase our perception of its brightness by aid of an instrument, 
which receives a beam four feet in diameter, and concentrates it 
upon a single point of the retina. Since the areas or sections 
of the two beams are as the squares of their diameters, the 
brightness of the star would be increased by the telescope in the 
proportion of 48 2 : 0.2 2 or 57,600 times, if the speculum were 
perfectly reflective. But only about § of the light is reflected 
from the speculum, and of this, about T V more is lost in trans- 
mission through two lenses ; the illuminating power is therefore 
0.6x57,600=34,560. A single quotation from one of Sir Wm. 
HerscheFs papers will illustrate the effect of a concentration of 
light so enormous. " I remember that after a considerable sweep 
with the 40-ft. instrument, the appearance of Sirius announced 
itself at a great distance, like the dawn of morning, and came 
on by degrees, till this brilliant star at last entered the field of 
the telescope with all the splendor of the rising sun, and forced 
me to take the eye from that beautiful sight." — Phil. Trans, vol. 
xc. p. 54. 

12. Atmospheric obstacles, and other causes of limitation to mag- 
nifying power. The four surfaces of an achromatic object-glass, 
or the single one of a speculum, cannot be wrought to absolute 
perfection, or if they could, they could not remain so, but would 
oscillate on either side of it by change of temperature. As the 
effects of such imperfection are magnified by the whole optical 
power of the instrument, this cause alone prevents the applica- 
tion of glasses of extremely short foci. 

Again, as the power is increased, the field is very much di- 
minished, and the diurnal motion of the earth is magnified in the 
same ratio. The field with 600 or 1000 will little more than 
comprise one of the larger planets, and a star vanishes at the 
edge of the field in 2 or 3 seconds. It requires much skill in 
the observer to keep the object in the field at all, and more to fol- 
low the star, manage the focal adjustment, and exercise severe 
scrutiny, all at once. 

But the chief bar to magnifying power is in the atmosphere. 
The air is seldom so well balanced, as to be without contrary 



THE TELESCOPE. 11 

currents and motions, which produce slight transitory undula- 
tions and variations of density. In a section of the broad beam 
of light which is to enter the telescope, there is room for con- 
siderable momentary differences of density, and consequently 
of refraction. Where the rays before incidence are thus irregu- 
larly deflected from perfect parallelism with each other, and 
quivering or vibrating about a mean state, the point of light 
which they form after reflection must be troubled and confused. 
The star in the field of view is in constant and rapid agitation, 
like an object seen through smoke or heated air, or the image of 
the sun on gently rippling water. According to different cir- 
cumstances of atmospheric disturbance, the appearances of the 
star to an attentive observer, are very different. Sometimes the 
individuals of a close double star twirl round each other, altering 
much their angle of position ; sometimes it appears like a drop 
of agitated mercury, — very frequently writhing and convulsed, 
breaking in pieces and reuniting, — occasionally heaving gently 
like the sun reflected on a calm swell ; — and the rarest of all 
states is that of perfect rest. These motions are minute, but are 
excessively troublesome, and in the majority of evenings prevent 
the application of very high powers, or the separation of very 
close and difficult stars. An excellent telescope might very easily 
be condemned for supposed want of optical capacity on an un- 
favorable evening. Its merits should not therefore be judged of 
too hastily, nor until several evenings' patient trial. And unless 
the star is free from any rapid quivering or agitation, it cannot 
be judged whether the atmosphere or the telescope is most in fault. 

13. There are many causes of disturbed equilibrium which it 
is in the power of the observer to remove. A telescope pointed 
out of a window, especially in winter, or over houses and walls 
heated by the sun during the day, will exhibit objects in a very 
inferior manner ; and, when brought out of doors, it needs a 
quarter or half an hour to acquire perfectly the temperature of 
the surrounding air. It is difficult to decide upon the atmospheric 
circumstances most favorable to perfect action in telescopes. As 
a general rule, it may be observed, that stars seldom appear well 
in clear frosty nights, or when they twinkle much to the naked 
eye. Sir Wm. Herschel remarks that " when the outsides of our 



12 THE TELESCOPE. 

telescopes are dropping with moisture discharged from the at- 
mosphere, there are now and then favorable hours in which it is 
hardly possible to put a limit to magnifying power ;" and it is 
not uncommon to find that a slight haze improves very much the 
definition of close double stars. 

As the aperture of a telescope is increased, the breadth of the 
beam of light which traverses the atmosphere becomes larger, 
and affords room for greater inequalities of density. The at- 
mosphere thus puts a speedy bound to all human efforts made 
within a medium so fickle, and must frustrate attempts to em- 
ploy telescopes of enormous magnitude. The 40-ft. telescope of 
Sir Wm. Herschel was nearly useless from this cause alone ; for 
there were scarcely 100 hours during a year, in which it could 
be used to any advantage. 

14. The observer will be a little surprised to find that stars 
have neat, well defined discs, especially since it is customary in 
philosophical treatises, to say that they are absolute points, with- 
out apparent size. This very common remark is true in this, 
that no magnifying can give the stars a real diameter. But 
a Lyrae, or any other bright star, appears in the telescope with 
a well defined and sharply terminated disc, of definite and sen- 
sible diameter, as certainly as Jupiter or Saturn. The differ- 
ence between them is not in appearance, but in fact. The discs 
of the stars are spurious, and are due to irradiation, which al- 
ways increases the diameters of intensely bright objects, — a phe- 
nomenon probably due to retinal sympathy. They are very 
small, and often confused amidst the aberration caused by im- 
perfections of figure, tremors, and their own dazzling brightness. 
In the best telescopes, on the finest nights, and with high powers, 
they are most neatly defined, and beautifully seen ; — and are then 
steadily surrounded with a number of slender circular rings of 
light, concentric with each other and with the star. Stars of the 
first magnitude have the largest diameter, and below a certain 
brightness this phenomenon ceases on account of growing indis- 
tinctness and want of stimulating power. 

15. Comparison of Refractors and Reflectors. 

We are now prepared to form some comparison between the 



THE TELESCOPE. 13 

respective advantages of achromatics and reflectors, of which 
latter we will take the Herschelian as the simplest form. The 
achromatic is less subject to atmospheric disturbances, and per- 
forms more uniformly at all times and under all circumstances ; 
while the speculum of the reflecting telescope is often liable to 
tremors, and requires very perfect equilibration of temperature. 
On the other hand, the discs of stars appear smaller in reflectors, 
and are therefore more easily separated, when close double. 
Reflectors too are in their nature perfectly achromatic, and color 
can only arise from the use of deep eye-lenses. They admit of 
being mounted more steadily than refractors ; for the object-metal 
of a reflector is below the eye of the observer, resting nearly or 
quite on the ground, — while the object-glass of the other is high 
above the observer's eye in the air, and very frequently set in 
vibration by every breath of wind. Again, vision is more easy 
in reflectors ; for the eye in the Herschelian looks downward, 
as in reading a book, and in the Newtonian, it looks direct- 
ly forward horizontally ; but with achromatics, especially at 
high altitudes, the head must be bent backward and held in a 
posture very tiresome and inconvenient, unless supported. 

The apertures of achromatics are usually less in proportion to 
their focal lengths, than those of reflectors. We will adduce a 
few instances by referring to several telescopes : 

Achromatics. 
Sir J. Herschel's 7-ft. equatorial, 5 inches aperture. 

Several 10-ft., by G. Dollond, 5 " 

Reflectors. 
The favorite 7-ft. of Sir Win. Herschel, Q\ inches aperture. 
40-ft. " 48 " 

These, and many other instances, show that the proportion 
found best in practice is for reflectors about 1 foot of focal length 
for every inch of aperture, and for achromatics between 1 and 
2 feet of focal length for the same aperture. But since glass 
transmits much more light than polished metal reflects, achro- 
matics, though of smaller aperture than reflectors of the same 
focal length, are nearly equal in light and power, and we shall 
regard them as such in speaking of the test-objects. 



14 THE TELESCOPE. 

16. On the test-objects of telescopic power and light. 

The usual objects of telescopic examination, especially the test- 
objects, will afford to the observer occupation, rich in interest 
and novelty. The solar spots, irregular dark forms, surrounded 
with their umbrae, — the bright ridges or faculse, interwoven like 
the vessels of a leaf, — and the mottled appearance of the whole 
disc of the sun, are easily seen in a 5-ft. telescope, with a power 
of 100, provided the dark glass be good. Mercury and Venus 
require magnifying power, diminution of aperture, or slight haze, 
to take off their dazzling brilliance, and are usually very tremu- 
lous. The phases of the former may be seen indistinctly, and 
those of the latter with beautiful precision and clearness under 
favorable circumstances. The variegated appearance of the 
moon, her diversity of light and shade, wide level plains, annular 
mountains, and lofty ridges, render her in any telescope the most 
striking and magnificent of astronomical objects. No employ- 
ment can be more interesting to the amateur astronomer than to 
watch from night to night the boundary of light and shade, as it 
passes over the lunar disc, — to see the mountains first appearing 
as separate fragments of light, soon joining and advancing into 
the bright portion, but still casting their long shadows back into 
the darkness, — to see these shadows separating at last from the 
dark portion, and gradually shortening as the long lunar day ad- 
vances. The strong contrasts of light and shade, and dazzling 
brightness of the lunar disc, are apt to surprise and confuse the 
observer ; but a little familiarity with its appearance and changes 
will very soon accustom him to judge as correctly of the ine- 
qualities of her surface, as of the irregularities of the ground 
beneath him, when lighted by strong sunlight. A power of 100 
or more will show that the circular edge of the moon's bright 
limb is slightly irregular, and-broken by mountains and valleys, 
seen in profile. This beautiful object is best exhibited in her first 
and last quarters, and is least interesting when full. 

The moons of Jupiter are within the reach of any telescope 
above 1 ft. in focal length. His two principal belts are rather 
more difficult objects, and to inspect them minutely requires a 
power of more than 100, and an aperture of several inches. 
With a good 7-ft. reflector or achromatic, the transits of satel- 
lites and their shadows over his disc may be observed with ease. 



THE TELESCOPE. 15 

The ring of Saturn is easily seen with a 2 or 3-ft. achromatic. 
The belts upon his body require light as well as distinctness ; but 
a power of 100 on a 5-ft. reflector should show them to advan- 
tage. The black division which separates the ring into two is a 
severe test for good telescopes. It can be seen in excellent tele- 
scopes of 4 or 5 feet focal length, or even less, but the powers 
of a 7-ffc. are requisite to give a satisfactory view of it. Even 
with an instrument of this magnitude, it is very rare to see the 
ring divided any where but at the two extremities of _^-^_ 
the oval, as in the annexed cut. To trace the divi- ((CL_Jt})) 
sion where narrowed by the effect of projection, and 
nearly throughout the visible portion of the ring, requires the 
most favorable night, perfect steadiness in the instrument and 
great excellence of figure, a practised eye, and the planet near 
the meridian. The telescope that has exhibited the division well 
at the extremities of the ring, has passed a very severe test, and 
is competent to deal with the greater part of Herschel's first class, 
or the closest of double stars. 

17. The double and triple stars present an endless variety of 
the most delicate objects to the attention of the observer. They 
are — first, of all degrees of proximity ; — secondly, of all grades 
of inequality. Many are farther apart than the diameter of Ju- 
piter, (from 30" to 40",) and some are closer than the breadth 
of the division in Saturn's ring, which is less than 1". The 
black interval between the two stars of Castor (fig. 1.) is about 
once the diameter of the larger, or l£ times that of the smaller 
star ; those of y Leonis (fig. 2.) very nearly touch one another ; 
and the two nearest of £ Cancri (fig. G.) are seen nearly or quite 
in contact. A closer approximation makes the discs overlap one 
another, appearing like a double-headed shot, as y Virginis in 
1838 (fig. 9.), or if very unequal, wedge-shaped, like X Ophiuchi 
(fig. 4.) If still closer, two equal stars appear merely a slight 
elongation of one, as y Virginis in 1837 (fig. 9.), and a faint 
companion only distorts the disc of the larger star. A star, how- 
ever, may present difficulties, not from the closeness, but from the 
faintness of its companion. And the observer, by visiting in 
succession the small stars attendant on Polaris, a Lyras, and y 



16 



THE TELESCOPE. 



Coronae,* will gradually descend to a degree of faintness, which 
requires long attention to be visible at all. 

Double stars vary in still another particular — viz. color. A 
few offer splendid contrasts, among which are y Andromedae, red 
and green, — (3 Cygni and s Bootis, in both of which the smaller 
is a fine blue star. The larger star is never blue or green; the 
smaller in these combinations may be either, and occasionally we 
find it of a fine purple. 

18. The following list of test-objects will enable the observer 
to choose such as will try the powers of his instrument to the 
utmost. 

A LIST OF TEST-OBJECTS. 
For an Achromatic, Newtonian, or Gregorian of 3-ft. or less. 



DEFINING POWER. 



Object. Dist. Mags 



/3 Cygni 

$ Urs. Maj. 
y Androm. 

y Arietis 
Castor 
a Piscium 
£ Aquarii 



34" 

14 
10 

9 

5 
5 

4 



3,4 
3,5 

4,4 
3,3i 
4,4 
5,5 



Remarks. 



L. yellow; S. fine 

blue. 
Coarse. 
Easy. S. beautiful 

green. 
Easy. 
Binary. ) 

> Close. 



Jupiter's moons and belts, — Saturn's ring. 



ILLUMINATING POWER. 



Object. 



/? Cephei 
77 Cass. 

I Bootis 
\ Librae 
Pleiades 

Prassepe 
1st Satellite 



Dist. Mags. 



13" 
10 



5,7 
5,8 



Remarks. 



Sm. star purple. Bi- 
nary. 
Binary. 
Binary. 

In Taurus. Very 
coarse cluster. 
In Cancer. Coarse cluster, 
of Saturn. 



For a 5-ft. Achromatic, or Herschelian. 



DEFINING POWER. 


ILLUMINATING POWER. 


Object. 


Dist. 


Mags. 


Remarks. 


Object. 


Dist. 


Mags. 


Remarks. 


y Aquarii 


4" 


5,5 


Very easy. 


| Libras 


7" 


5,9 


Triple and revolv- 


el Lyrse 
e' Lyras 


31 


6,7 
6,6 


) Beautiful double- 


Rigel 


9 


1,9 


ing. 


\ double. Both 


£ Bootis 


3 


3,8 




3 


) binary. 


Polaris 


15 


2,10 




11 Monoc. 


3,7 


7,8,8£ 


Qh. 20m. — 6° 55'. f 


39 Drac. 


3i 


5,10 










Easy triple. 


o- Coronas 


40 


7,13 


Quadruple. The 


£ Leonis 


3 


2.3,4 


Difficult. Binary. 








most distant. 


y Virginis 


Var. 


3,3 


Rapid binary. 


% Persei 


Two very rich clusters, crowd- 


£ Libras 


H 


5,5 


Very difficult. Bi- 




ed with stars. 








nary. 


Saturn's be. 


ts and 3 satellites. 


Division of ' 


Saturn's ring at the extremities. 







* This star is also close double, and has been a standard test-object for telescopes, 
but of late has closed so as to be inseparable by almost every telescope in the world. 

t The places of objects not in the small " Maps of the Society for the Diffusion of 
Useful Knowledge," are designated by their right ascensions and declinations for the 
vear 1830. 



THE TELESCOPE. 



17 



For a 1-ft. Achromatic, or Herschelian. 



DEFINING POWER. 


ILLUMINATING POWER. 


Object. 


Dist. 


Mags. 


Remarks. 


Object. 


Dist. 


Mags. 


Remarks. 


$ Bootis 


1" 


5,5 




Polaris 


15" 


2,10 


Easy. 


$ Cancri 


5,1 


6,7,7 


8h. 2.1m. + 18° 10'. 


a Lyrae 


45 


1,12 


Difficult. 








Triple. Rapid re- 


v Coronae 


40 


6,13 


15ft. 16.1???.-}-30°56'. 








volution. 








Very difficult. 


a Coronas 


Var. 


7,9 


16ft. 7.5m.-j-34° 20'. 


Debilissi- ^ 








\ Ophiuchi 


1 


4,6 


Rapid binary. 
Binary. Wedge-sha- 
ped. 


ma inter I 
e l snd e 2 [ 
Lyrae J 


50 


13,15 


Very difficult. 


7r Aquilae 


1 


7,7 
8,9 


19/j.40.7m.4-ll°24'. 
Ih. 31.1m. 4-5° 37'. 


Messier 5 
" 13 






) Resolvable ; 


Can.Min.31 






> glimpses of innu- 








In the field with 








) merable stars. 








and following Pro- 


5 satellites of Saturn. 








cyon. 





For a 10-ft. Achromatic, or Herschelian. 



DEFINING POWER. 


ILLUMINATING POWER. 


Object. 


Dist. 


Mags. 


Remarks. 


Object. 


Dist. 


Mags. 


Remarks. 


36 Androm. 
P. XIII. 127 

£ Arietis 
4 Aquarii 


It 

1 
1 

1 


7,7 
9,10 

7,7i 
7,7i 

8.8 


Oft. 45.6m.-f 22° 41'. 

13ft. 25.4m.+0° 35'. 
In the field with 
and north of £ Vir- 
ginis. Binary. 

20ft. 42.1m. -60 17'. 
Power 800. Re- 
quires light. 

16ft. 55.5m.-f 65° 19'. 


y Crateris 
t Orionis 
Virginis 
5 Cygni 

Messier 5 
11 13 


3" 

18 
4 
2 


4,13 
4,14 
5,14 
3,10 


Very difficult. 

Needs defining 

power. 
) Partially resolved 
5 into stars. 






20 * Drac. 









For a 14-ft. Achromatic, or Herschelian. 



DEFINING POWER. 


ILLUMINATING POWER. 




Object. 


Dist. 


Mags. 


Remarks. 




/? Aquarii 


20 


3,15 






t Bootis 


20 


4,16 




The same as for a 10-ft. telescope. 


<r Coronae 


20 


7,16 


Nearest of the two 
faint stars. 






"\ Crowded with countless 




Messier 5 


1 stars, which run up to a 




" 13 


f confused blaze of light in 
) the centre. 







For a 20-ft. Herschelian. 



DEFINING POWER. 


ILLUMINATING POWER. 




Object. 


Dist. 


Mags. 


Remarks. 


The same as for a 10-ft. telescope. 


| Pegasi 
a 2 Cancri 

a 2 Capric. 


11 
10 

8 


5,16 

4.5,16 

3,16 


The nearest of two. 



18 THE TELESCOPE. 

These tests of distinctness and light are such as ought to be 
within the reach of very good telescopes of the focal lengths 
assigned, and of the apertures usually corresponding to them. 
Occasionally an object-glass or speculum is fortunately wrought 
to such perfection, that it may resolve nearly all of the tests 
which have been recommended for a larger telescope. The ob- 
server, however, with an instrument of but common excellence, 
can scarcely hope to do more than separate the tests appropriate 
to its size ; and indeed, only the easiest of these, while unprac- 
tised in observation, and unfamiliar with atmospheric obstacles 
to perfect vision.* 

19. A few of the more remarkable combinations of double 
and multiple stars are represented in figs. 1 — 9. 

Figs. 1 and 2. — Castor and 7 Leonis with a power of 300. 
Standard and beautiful objects, on account of the brightness of 
the individuals. Both slow binary. 

Figs. 3 and 4. — 39 Draconis and X Ophiuchi with 300. The 
latter is binary, and so rapid in its revolution, that a few years 
will probably render it too close to be visible ; it can now be 
seen only wedge-shaped. 39 Draconis consists of 2 stars, the 
larger of which is the double star. 

Figs. 5 and 6. — 1 1 Monocerotis and % Cancri with 300. The 
former is stationary. The distant star of £ Cancri is in very 
slow motion ; and the two nearest are in rapid rotation, f Li- 
brae, and probably 12 Lyncis (6 h - 30 m . +59° 37') are also triple 
stars in revolution around a common centre of gravity. 

* The actual performance of an excellent 7^-ft. Herschelian of Mr. Holcorabe's 
manufacture will afford the student an illustration of the advantage of reference to 
test-objects. With the instrument referred to, | Libras and $ Bootis have been often 
and easily separated, and £ Cancri well elongated. A rare night early in 1838, show- 
ed y Virginis and A Ophiuchi notched on either side, it Aquila? was pronounced a 
very easy star, and Saturn's ring was seen double nearly throughout its visible por- 
tion. 36 Andromedaa is also within the reach of the instrument. " Debilissima" was 
first looked for through a very thin haze, but was distinctly seen with 5 inches aper- 
ture by glimpses, and correctly figured. And since on referring to European observa- 
tions, we find that few or none of their telescopes of equal focal length are competent 
to resolve closer test-objects than these, we are thus enabled to decide on the great 
excellence of Mr. Holcombe's instrument. It may be of use to the inexperienced ob- 
server to add, that at first it was thought no small achievement to separate y Leonis 
with this telescope, and £ Libras could not be seen at all. 



THE TELESCOPE. 19 

Fig. 7. An accurate representation of the fine double-double 
star s 1 and s 2 Lyrse with 150. The unequal set is s 1 , the equal is 
s 2 , and both are slow binary stars ; a star of the 10th mag. fol- 
lows on the right hand. The faint " Debilissima" is between, 
the uppermost of the two being the brightest, and both necessa- 
rily much exaggerated. 

Fig. 8. a Orionis with a power of 80. Composed of 7 stars 
easily visible in a 3 or 5-ft. telescope. 

Fig. 9. 7 Virginis with 300. In June 1836, it passed its pe- 
rihelion ; then, and in 1837, scarce a telescope in the world could 
show it otherwise than round, and none more than a trifle elon- 
gated; in Jan. 1838, the elongation might be seen notched on 
each side ; in Jan. 1839, there was a hair-breadth division; and 
now in 1840, it is fairly and easily separated with a large tele- 
scope. Its appearance for 1845, 1850, and 1860, are also given, 
and the several phases when combined, plainly indicate that the 
elliptical orbit annexed to the figure, is described by one star 
relatively around the other. These phases are taken from the 
distances and positions of the table in Art. 29. 

These figures represent the stars as they are seen in an achro- 
matic telescope when on the meridian, and by holding them be- 
fore a looking-glass, they will appear as in a Herschelian. The 
letters n,f, s, p, signify respectively " north," " following," " south," 
" preceding," and mark out those directions in the heavens. The 
revolving stars naturally excite the most interest, and these rep- 
resentations therefore, (with the exception of 39 Draconis, 11 
Monocerotis, and <f Orionis.) are chosen from that class, and ex- 
hibit them in their true positions for the year 1840. 

20. The nebulas and clusters of stars are in a large telescope 
objects of uncommon interest, because different from any thing 
we see with the naked eye. Of these the Nebula in Orion, of 
an irregular shape, and containing a minute trapezium of stars, 
is the brightest. It has a nebulous star in the same field. The 
Nebula in Andromeda (M. 31) is a very long ellipse, fading im- 
perceptibly away from a bright centre ; there is a nebulous star 
(M. 32) almost in the same field with it, and a faint nebula at a 
little greater distance. These two bright nebulas are well seen in 
achromatics of 3 and 5-ft., and are splendid objects in large tele- 



20 THE TELESCOPE. 

scopes. There is a pretty bright nebula in Sagitta, (19 h - 52.2 m . 
+22° 16',) which in a 7 or 10-ft. may be seen as a double-headed 
shot, or double nebula not quite separated. Between (3 and y 
Lyrae is a wonderful annular nebula (M. 57 ;) it requires a good 
eye to distinguish the " hole in it" in a 7-ft. reflector. Through 
the double star 52 k Cygni (20 h - 38.9 m . +30° 7',) passes an ex- 
ceedingly faint, forked ray of nebulosity, which can but just be 
seen in a 10 or 12-ft. instrument. 

In Cancer, the naked eye may easily see a nebulous appear- 
ance, like a comet, called Praesepe, (8 h - 30.4 m . +20° 34',) which a 
3-ft. telescope shows to be a brilliant cluster of large scattered 
stars. Midway between Cassiopeia and Perseus, in the sword- 
handle of the latter, a hazy star, x Persei (2 h - 7— 10 m . +56° 22') is 
faintly visible to the eye, which a 5 or 7-ft. telescope exhibits as 
two clusters of crowded stars, each filling the field — a glorious 
object. These, however, are but coarsely scattered collections 
when compared to Messier's 13th (16 h 35.7 m . +36° 48') and 5th, 
which in a 7-ft. are nebulous, with the addition of a few faint 
glimpses of stars ; a 10-ft. shows many faint stars, but it requires 
at least a 14-ft. to resolve all the nebulosity into stars, and ex- 
hibit them as dense swarms, running up to a blaze of light in the 
centre. M. 13 is between tj and % Herculis, and M. 5 is almost 
or quite in the same field with 5 Serpentis. 

21. How to find objects for examination. 

A familiarity with the geography of the heavens is so ne- 
cessary for the observation of such objects as we have named, 
that in no way can it be acquired more rapidly than by looking 
for faint stars with the telescope. A good map of the heavens is 
here an indispensable auxiliary. The best of the kind easily ob- 
tained are those of " The Society for the Diffusion of Useful 
Knowledge, London." The larger set are expensive, but the 
price of the smaller does not exceed $2. 

22. Stars are numbered on maps and in catalogues in several 
different ways. Thus — a Orionis — 58 Orionis — P. V. 268 are 
synonyms for the bright star Betelgeux. 58 Orionis means the 
58th star in order of right ascension, of those which Flamsteed 
numbered in Orion, and P. V. 268 refers to the 268th star in or- 



THE TELESCOPE. 



21 



der of right ascension between 4 h and 5 h R. A. in Piazzi's cata- 
logue. Double stars and nebula? are referred to in catalogues by 
abbreviations like the following : H. II. 79, is the 79th double 
star or nebula of Sir Wm, Herschel's second class ; h. 1239, de- 
notes a double star or nebula of that number in the younger 
Herschel's catalogues; M. 13, is the 13th nebula of Messier's 
catalogue, &c. 



23. The points of the compass in the heavens are — north, — 
following, — south, — preceding, — denoted by their initial letters. 
A star that precedes another arrives at the meridian first, and its 
right ascension is therefore less than that of the other. The pre- 
ceding point of the field of view is always that towards which 
a star moves by diurnal motion ; and the observer, by noticing 
this circumstance, is at no loss to conceive of the other car- 
dinal points in his field. For in an achromatic, the points n, f 
s, p, follow the circumference of the field in a direction contrary 
to that in which the hands of a watch revolve ; in a Herscheiian, 
they follow in the opposite direction, or with the hand of a watch. 
The following cut exhibits the different positions of these points 
when the star moves horizontally, or is on the meridian. 



In 



ACHROMATIC. 



HERSCHELIAN. 
S 




South of the zenith. 



North of the zenith, above the pole. 



North of the zenith, below the pole. 



It is to be noticed that an achromatic inverts entirely, both up 
and down, and right and left ; while in the Herscheiian objects 



22 THE TELESCOPE. 

are only inverted up and down, as when seen in a still lake. 
This is because the mirror, at the same time that it inverts like an 
achromatic, reverses right and left as in a common looking-glass, 
thus destroying the effect in a horizontal direction. In a New- 
tonian, because of the second reflection, objects appear as in an 
achromatic. / 

24. Suppose the object to be looked for is put down on the map, 
and visible to the naked eye, for instance, 39 Draconis. A kind 
of trianguiation may be carried on by means of the map from 
the large and well-known stars in the neighborhood down to the 
point required. The star, when found, should be so carefully 
compared with the surrounding objects by means of the map, 
(especially by ranging it with stars nearly or quite in the same 
line,) that no doubt may remain of its identity. For without 
much care and precision, fancied resemblances of bearing and 
distance may be mistaken for real ones, and the observer may be 
mortified with the discovery, that he has been exercising much 
needless scrutiny and attention upon the wrong star. If the ob- 
ject is invisible to the naked eye, the exact point among the stars 
towards which the telescope is to be directed, may be determined 
in the same way. If not recorded on the map, a pencil dot may 
be made to mark its place, when its right ascension and declina- 
tion are given. 

When the star, or the point where it should be seen, if visible, 
is thus determined by the eye, — direct the tube of the telescope 
or finder as nearly as possible to the spot ; and make the image 
of the cross-wires, viewed with one eye, approach the desired 
star or point, as seen with the other. It is usually best in prac- 
tice, to bring the nearest bright star on the cross-wires, and 
then move in the direction of the object, still employing binocu- 
lar vision. By remembering that the finder inverts, and noticing 
the scale of its enlargement, it soon becomes easy to trace the 
inverted and magnified but still similar configurations of the 
finder, proceed from star to star, and thus arrive at the object 
in the finder, though invisible to the naked eye. 

25. Other modes of finding an object, when invisible to the 
naked eye, will be but hinted at. An object invisible even in 



THE TELESCOPE. 23 

the finder, may often be detected by ranging in its vicinity, with 
one observer at the finder, and another watching the field of 
view ; the annular nebula in Lyra is thus found without much 
difficulty. If an invisible object follows a visible star in the same 
parallel, and is less than a diameter of the field of vision north 
or south of that star, it may be found with the aid of a common 
watch, by allowing the diurnal motion of the heavens to sweep 
past the field of view the partial zone that intervenes between 
the star of departure and the object to be found. If the tele- 
scope can be so placed in the meridian as to rise and fall through 
a small zone with tolerable correctness, (for which purpose such 
simple methods as suspension of the upper end by a cord and 
pulley, or a bearing of the same against a wooden bar truly 
upright, will avail,) the faintest objects within the reach of the 
telescope may be found without the aid of a map by their right 
ascensions and declinations, taking care to note the passages and 
declinations of a sufficient number of known stars to furnish 
standards of comparison. A common watch will answer to 
measure right ascensions, and any good scale of equal parts will 
serve for the small zone of declination. 

26. How to examine a difficult object to advantage. 

Suppose the object is a close double star, which has been found 
with a low power. Put in as high a power as the object will 
easily bear, and adjust for distinct vision. It is easy to hit a point 
of tolerably distinct vision by turning one way and the other, 
for growing indistinctness on either side shows the necessity of 
turning back. But on a delicate object the most perfect vision 
of the object can only be obtained by adjusting this point with 
the utmost nicety and repeatedly, and the focal point that is best 
suited for one eye will not answer for another. A near-sighted 
eye requires that the eye-piece be pushed in a little farther than 
usual. The object should be kept in the middle of the field, or 
suffered to cross the field diametrically several times. If the 
telescope is really excellent, and the object looked at not a very 
difficult one, few evenings occur in which, however disturbed the 
star may be, there are not momentary pauses and quiescences, 
during which a quick eye may catch a glimpse of neat and per- 
fect separation between two well defined discs. A few such 



24 THE TELESCOPE. 

glimpses will settle the point. If the night is a rare one, the 
most difficult objects may be attempted, and these should be re- 
served for such occasions, for it is useless to attempt to deal with 
them at ordinary times. 

If the object is the very faint companion of a star, or the most 
diffuse parts of a nebula, a surprising advantage is usually gain- 
ed, by directing the eye to a part of the field at some little dis- 
tance. Besides indirect vision, a careful seclusion and rest of 
the eye from all light, even from that of the sky, for a short time, 
increases its susceptibility to the impressions of light. A mode- 
rately high power is sometimes needed upon faint stars to render 
them visible, although usually low powers are considered brighter. 
In all cases, whether the object be close or faint, different eye- 
pieces and different single glasses should be tried successively, to 
ascertain with what combination the effect is the best. 

27. To remedy the defects of a telescope. 

The outer border of an object-glass or speculum is frequently 
imperfect, on account of the difficulty o£ figuring and polishing 
surfaces true to the very edges. A great advantage in defining 
power is frequently gained by a diaphragm, or a succession of 
diaphragms, that shall contract the aperture by little and little. 

All optic glasses, when dirty or dusty, should be wiped care- 
fully with chamois leather, or clean soft cotton, free from dust, — 
never with silk, which will in time wear many scratches. But as 
a general rule, it is best to wipe the glasses as seldom as possible, 
and then lightly and quickly. Specks of dust interfere very 
much with perfect vision, especially on eye-glasses, and it is there- 
fore almost needless to caution the observer to keep them con- 
stantly and closely covered, when not in use. The speculum of 
a reflector is more liable to become tarnished and dirty than an 
object-glass ; but since it is more easily scratched, it is best to wait 
until the evil becomes very troublesome. A closely adhering 
film of dirt, and even an incipient oxidation, may be removed by 
spreading over the surface a drop or two of oil of turpentine, 
and, when it is nearly dry and hard, applying a piece of soft 
leather, with a firm hard pressure. The strong adhesion of the 
oil will bring off with itself all that can be safely removed from 
the surface of the speculum. 



THE TELESCOPE. 25 

The action of a telescope that is unsteady in windy weather 
may usually be improved by bearing the farther end of the tube 
against some steadily yielding support, as a bar of wood leaning 
against the side of the telescope, or a double-folding window- 
shutter. 

28. We will conclude the subject by pointing out to the ob- 
server the different ways, in which he may employ himself agree- 
ably and usefully with the aid of the telescope. 

There is enough of strong interest and deep instruction in sur- 
veying the more secret of the wonders of the sidereal heavens, 
to render the use of the telescope, with no other object than per- 
sonal information, an occupation elevating to the mind, no less 
than it is novel and delightful. And one who thus seeks for him- 
self and his friends, a more intimate acquaintance with the grand- 
est works of nature, than is commonly obtained in the progress 
of education, cannot be said to spend the fraction of time which 
he devotes to these studies otherwise than most usefully and pro- 
fitably. At the same time, there are few observers, who, if in 
addition to these sources of gratification, they could be conscious 
of adding a tribute, however humble, to the sum of human 
knowledge, and of advancing a little a science, whose long and 
splendid history, and present greatness, is written in the succes- 
sive contributions of individuals, would not feel a more keen in- 
terest and animation in the pursuit. A few of the more eligible 
modes of employing the telescope to advantage, will therefore 
be suggested. 

29. There have been multitudes of those curious systems, call- 
ed double and triple stars, observed, carefully registered in dis- 
tance and position, and inserted in catalogues. A considerable 
part of them have been reviewed, and material, sometimes very 
considerable changes of angle and distance in a very few have 
at once detected a rapid revolution. Great numbers have never 
been examined but once ; and among these some revolving or 
binary stars doubtless lie concealed for want of a second obser- 
vation. The angle of position with the meridian is recorded in 
catalogues in two different ways ; the old method reckons it 
from 0° to 90° in each quadrant from the parallel towards the 

4 



26 



THE TELESCOPE. 



meridian ; while that more recently adopted reckons from the 
north point, around the circle in the direction n, f, s, p. Thus 
10° s p corresponds to 260° of the new nomenclature. The ob- 
server may sweep a small zone of the meridian in the way re- 
commended in Art. 25, and examining each faint and very close 
double star that has not been repeatedly observed before at very 
distant intervals of time, record its approximate distance and 
angle of position. The diagrams in Art. 23, will readily suggest 
the mode of estimating the latter, and the insertion of corre- 
sponding cross-lines in the field of view faintly illuminated will 
much aid the judgment. Any star that has varied its angle of 
position by 10° or more, would in this way be instantly detected 
by a careful observer. And the demonstration of the compara- 
tive fixity of the remainder would be of no inconsiderable im- 
portance. 

Thus the observer could not mistake the changes either in dis- 
tance or position, which have occurred and which will occur in 
y Virginis, as represented in fig. 9. That he may avail himself 
of its rapid motion to try the accuracy of his estimations from 
year to year, we subjoin a table of the successive meridional ap- 
pearances of this interesting system until 1860: 

y Virginis. 



Date. 


Distance. 


Position. 


Date. 


Distance. 


Position. 


Date. 


Distance. 


Position. 


1837.0 

1838.0 
1839.0 


0."5 

0. 8 

1. 1 


109° 
60 
41 


1840.0 
1841.0 
1842.0 


1".4 
1. 6 
1. 8 


31° 

25 

20 


1845.0 
1850.0 
1860.0 


2".3 

3. 1 

4. 3 


10° 

1 

350 



tf Coronae, another of our test-objects, is rapidly increasing its 
distance in much the same way, and from nearly the same epoch 
of perihelion passage. 

A good micrometer attached to a telescope will furnish em- 
ployment enough to the most active astronomer. It will enable 
him to detect far slighter changes in distance and position than 
mere estimation can do, — to confirm old determinations, — and 
to settle the elements of new objects. (See Art. 33, on " the 
Micrometer.") 



30. Again, there are probably many telescopic comets roam- 
ing abroad in different parts of the heavens, especially in that 



THE TELESCOPE. 27 

part of it which is nearly lost in twilight. A telescope of large 
aperture and great light, with very low power and a wide field 
of a degree or two, or even more, (aptly termed a sweeper) is 
admirably adapted to explore carefully the twilight sky, and 
keep watch for comets. Any nebulous appearance is quickly 
detected, and its motion and situation will show very soon 
whether it is a nebula or comet. The announcement of the 
discovery of such a body is instantly followed up by careful ob- 
servations upon it at the principal European observatories. 

31. Eclipses of the sun and moon, occultations of stars by the 
moon, the eclipses of Jupiter's satellites, &c, are events of much 
interest, and good observations of the exact moments of their 
occurrence are always esteemed important additions to practical 
science. If the observer has access to a transit instrument, or 
sextant, his telescope will afford very useful occupation in such 
kind of observations. An immersion or emersion of a bright 
star at the moon's dark limb is a beautiful sight, often absolutely 
startling, for the star flashes into view or vanishes from sight in 
dark space without a moment's warning, and as far as sense can 
discern, instantly. The time can usually be noted to the accu- 
racy of the fraction of a second at the dark limb, but is more 
uncertain at the bright limb, especially when the star occulted is 
below the 3rd magnitude. But to be well prepared for pheno- 
mena of so sudden occurrence, it is necessary to know before- 
hand the approximate time of immersion and emersion, to the 
nearest minute or two, and the angles from the moon's vertex 
reckoned around her circumference, at which the star will disap- 
pear and reappear. Directions for calculating these roughly 
will be given in the chapter on " Eclipses and Occultations." 
The observer, with such preparation, has only to count from the 
clock for two or three minutes before the time appointed, with 
his eye on that part of the moon's edge indicated by the calcu- 
lated angle, and very nearly at the expected instant and place, 
the star will appear or disappear, as the case may be. The chap- 
ter on " the Transit Instrument" will direct him how to conclude 
the true time of the observations from the recorded instants, and 
he will then have the means of determining his longitude, if he 
chooses, for which purpose he may consult the chapter on that 



28 THE MINOR ASTRONOMICAL INSTRUMENTS. 

subject. But, if he does not wish to carry his observations to 
their conclusions, their publication will place them in the hands 
of those, who will make the requisite use of them. 

But there are a number of highly interesting phenomena, 
whose laws are as yet little more than conjectural, connected with 
eclipses and occultations ; these need the light that can be thrown 
upon them only by observation, and will render it worth the ob- 
server's while to watch these phenomena closely, even if he has 
no knowledge of the true time. Thus, the first indentation of 
the moon in a solar eclipse is probably in many cases a deep, 
acute notch in the sun's edge, soon blending with the moon's 
limb ; and in an annular eclipse, the phenomena attendant, on the 
formation and rupture of the ring are beginning to excite much 
interest and careful attention. In occultations, some bright stars, 
and Aldebaran especially, have been seen in a number of cases 
to advance apparently upon the moon's bright or dark limb, and 
that to a very perceptible distance, before they vanished, — a point 
strongly demanding the notice of future observers. And it is de- 
sirable to know whether there is ever any perceptible distortion or 
diminution of brightness in a star or planet on approaching the 
moon's dark limb, or any appreciable time occupied in the act 
of disappearance. These, and a number of other particulars, 
which are not yet free from mystery and uncertainty, need very 
much the aid, which active and careful observers, with no other 
instruments than excellent telescopes, can bestow. 



CHAPTER II. 

OF THE MINOR ASTRONOMICAL INSTRUMENTS THE MICROMETER 

LEVEL, &C. 

32. There are a number of instruments and appendages of 
instruments, which, connected with and dependent on the larger 
and more important, will be occasionally referred to in our de- 
scriptions of their uses. To a cursory notice of the principal 
of these we shall devote the present chapter. 



THE MICROMETER. 29 

They may be classed as follows ; — first, those which serve to 
measure small celestial arcs, either directly in the field of view, 
or indirectly on the limb of an instrument. These are microme- 
ters under their varied forms, the vernier, the reading micro- 
scope, &c. 

Secondly, those which aid in determining fixed points of the 
sphere, or of the limb of an instrument, especially the zenith and 
horizontal points. These are the level, the plumb-line, the artifi- 
cial horizon, the floating collimator, &c. 

33. The micrometer is the most important of all those instru- 
ments which are to be used only in connection with others. Its 
object is to measure distances in the field of view. Suppose we 
insert a pair of movable wires exactly in the plane of the aerial 
image or focus of a telescope, and are enabled to measure their 
separation by a delicate scale outside of the tube ; we obviously 
have the means of accurately measuring the dimensions and rela- 
tive distances of the images of planets and stars. For the images 
of celestial objects and the movable spider-lines are equally in 
the focus of the eye-glass, and the eye refers both to the sphere 
of the heavens. The scale of the micrometer is, however, only 
a comparative one, and the value of any portion of it, as an inch, 
or fraction of an inch, must be found in seconds of space by 
measuring bodies of known diameter, or stars of known distance, 
or by noting the time in which a star of known rate of motion 
passes over a given interval.* 

The numerous varieties of micrometers are all calculated to 
measure the focal images of the heavenly bodies by reference in 
some way or other to a delicate scale, and may be classed under 
three principal species. Theirs* comprises those in which some 
simple form of scale is inserted in the field of view at the focal 
image ; the principal of which is the wire or spider's line mi- 
crometer. Cavallo's transparent scale of mother of pearl, reticu- 
lated diaphragms and networks of lines, and annular microme- 
ters belong to this class ; and it may be said to include also the 
cross-lines in the field of view of all fixed instruments. The 
second are double image micrometers, in which, by the division 

* See the method of finding " the equatorial interval" in the chapter on " the Tran- 
sit Instrument." 



30 THE MICROMETER. 

of some lens, mirror, or prism, two images are actually made of 
the same star in the plane of the focal image, and their separa- 
tion is measured by the separation of the halves of the lens or 
mirror, and an attached scale. The chief of these is Dollond's 
object-glass micrometer. The third are binocular micrometers, 
in which distances and diameters seen in the field of view with 
one eye, are compared with some species of scale seen outside 
of the telescope with the other. 

34. We shall take for an example the most common form, the 
wire or spider-line micrometer, and refer to fig. 10. in our brief 
description of its parts and their use. It consists of a closed 
shallow box \ or i inch thick in the direction of the axis of the 
telescope, 1 or 2 inches broad, and 3 or 4 long, with a divided 
screw head at each extremity. Its general appearance, when 
attached to a telescope, may be seen in fig. 11. If we should 
take out the eye-piece, and remove the cover, the appearance in 
fig. 1 0. would be presented : aaaa are the sides of the containing 
box, seen edgewise ; the two forks of brass, bbb, ccc slide one 
within the other and in opposite directions, and across them are 
respectively stretched the spider-lines, d and e. To them are 
firmly fastened the screws Jf, which, passing through the ends 
of the box, enter the nuts or divided heads gg. It is obvious, 
that whenever these latter are turned in the direction indicated 
by the figures on their circumferences, the forks b, c will be 
drawn outwards ; and on turning in the contrary direction, the 
springs hh tend to push the forks inwards, and thus prevent any 
shake or loss of motion in the screw. The screws have about 
100 threads to the inch, and one revolution of the divided head 
g therefore carries the line d over the T £oth of an inch ; but to 
such exquisite perfection has the cutting of these screws been 
carried, that by dividing the circumference of the nut g into 100 
parts, T £o of each thread, or to£ot of an inch may be perfectly 
reckoned. The field of view is oblong, and within it are seen 
the lines d, e, and on one side a notched scale of teeth corre- 
sponding in size to the threads of the screw ; every fifth one of 
these are cut deeper than the rest, and they are numbered from 
zero at the centre by tens in each direction. The spider-lines 
may be brought to coincide at zero, and even to glide by each 



THE VERNIER. 31 

other a little way, e passing very close under d. Suppose them 
both at coincidence at zero of the scale, and that it is required 
to measure the diameter of the sun ; turn either nut, as g, until 
d is drawn so far out, as to touch one limb, while e touches the 
other ; read off in the field of view how many notches have been 
passed over by the wire, and on the divided head against a fixed 
mark the fractional part. Thus, if d is between the 22nd and 
23rd notch in the field, and the diamond mark k stands at 72, the 
measure will be 22.72 revolutions of the head. This can be re- 
duced to seconds of space by knowing the value of each thread 
of the screw, (see Art. 33.) If e also has been moved in the 
opposite direction, the space passed over by that thread must be 
added to the other. 

When this apparatus is attached to, and carried round by, a cir- 
cle, divided into degrees and parts, it becomes a position microme- 
ter. By placing the fixed wire I so as to bisect both the stars to 
be measured, their angle of position with the meridian may be as- 
certained. Fig. 11. represents such an arrangement ; the divided 
circle a, and with it the box b and attached eye-piece, are carried 
around to any position desired by turning the milled head s, and 
the angle is read off by aid of the fixed vernier c. 

35. The vernier. — It is an undertaking as difficult as tedious 
to divide the limb of an instrument with refined accuracy. This 
renders it desirable to make as few divisions as possible, and de- 
scend to any further subdivision by some contrivance, which can 
be applied successively to different parts of the limb. Such are 
the vernier and reading microscope. 

The principle of the vernier is as follows : If a small arc of 
equal radius with the limb of an instrument, and sliding in coin- 
cidence with it, be divided so that any number n of its divisions 
shall correspond to n — 1 or n+1 on the limb, it will enable the 
observer to subdivide each division of the limb into n parts. 
Without going into details, it is easy to see in fig. 12., that if the 
20 divisions of the vernier aa are equal to 19 on the limb bb, and 
the 4th of the twenty exactly coincides with any one on the limb, 
the diamond mark must be -^o of the distance from c to d. The 
space c d, between the divisions 32° and 32° 20', is thus subdivided 
into 20 parts or minutes, and the reading of the limb is 32° 4'. 



32 THE LEVEL. 

The reading microscope is the application of a spider's line 
micrometer to a microscope, to measure minute distances on the 
limb. The image of the limb is received in the same plane with 
the spider-lines like that of a celestial object, and the fractional 
part of cd pointed out by the zero of the micrometer may thus 
be measured. The instrument is represented at the end of the 
arc in fig. 12. 

36. The level is the most important of those instruments, which 
serve to point out the zenith and horizontal points, and enable the 
observer to reckon distances from them. One of its most valu- 
able applications is to the transit instrument, and a description 
of its use in this connexion will readily suggest the modifications 
in its adaptation to circles, and fixed instruments generally. 
There are two very common forms of the level, — termed the 
hanging and riding levels ; these terms relate to the position of 
the instrument with respect to the axis of the transit, and sufficient- 
ly explain themselves. The tube of the level is nearly filled with 
alcohol and hermetically sealed, so as to include a bubble of air. 
It is ground within so that the upper side shall be very slightly 
convex upwards, and when nearly levelled, the bubble will of 
course rest at the highest point of the curve, or where a tangent 
to it is exactly horizontal. Fig. 13. represents a hanging level, 
suspended from the axis of the transit, and embracing each pivot 
in the way exhibited in fig. 14. Its use is to render the axis of 
the transit perfectly horizontal. How then are we enabled to 
effect this object ? 

If the bubble does not rest at the centre of its attached scale, 
but runs towards the pivot P', it is because the pivot P' is 
higher than P, or the arm of the level L'M' is shorter than LM ; 
for either of these circumstances, or both of them combined, will 
raise the end a above the opposite one b of the tube. It would 
seem at first sight difficult to distinguish between that portion of 
the effect due to the inclination of the axis P'P, and the differ- 
ence of the arms L'M', LM. But suppose we reverse the level, 
end for end. If the axis is really (though unknown to us) in a 
horizontal position, the bubble that before ran towards P' because 
L'M' was too short, will in the reversed position run towards 
P, — that is, will remain at the same absolute point of the scale as 



THE PLUMB-LINE. 33 

before. But if the pivot P' be higher than P, and the arms are 
equal, the bubble will still run towards P' after reversion, and of 
course will no longer retain the same place that it did in the first 
position of the transit, but one as far on the other side of the 
central zero ; although apparently the same as before, because 
the reversion of the level reverses also its scale. 

Remembering, therefore, that a difference in the lengths of the 
arms of the level has no tendency to change the real place of the 
bubble in the two positions, but that all change or difference of 
place is due to inclination of the axis, we have an easy means of 
rectifying such inclination. For if after reversion, the bubble 
should be found not at the same real point of the scale as before, 
but should be removed towards either pivot, that pivot is of 
course too high. It must be lowered until the bubble has gone 
back half way to the point it occupied in the first position, for 
it is plain that the first point is as much too far on one side as 
the second is on the other. If now in the two positions, it re- 
mains at the same division of the scale, the axis is truly horizon- 
tal ; otherwise, the error yet uncorrected must be diminished by 
a second trial. 

After the transit axis is carefully levelled, if the bubble does 
not rest at the centre, it may be made to do so by lengthening the 
arm, toward which it tends. Where there is no screw for this 
purpose, the adjustment can always be effected by scraping or 
filing a little the internal angle of the shorter arm. 

The place of the centre of the bubble on the scale may ea- 
sily be inferred from the readings at its two extremities, and is 
half way between them. 

37. The plumb-line is always perpendicular to the surface of 
still water, and therefore marks the observer's zenith. A single 
example will illustrate its astronomical utility. In the altitude 
and azimuth instrument, a plumb-line is attached to the vertical 
axis at its upper extremity, and hangs by its side, so as exactly 
to cover the image of a fine dot near the lower extremity of the 
axis, when viewed by a reading microscope. The test (and a 
very severe one) of the exact verticality of the axis, is the exact 
bisection of this dot by the plumb-line, while the axis is turned 
completely round in azimuth. 

5 



34 THE SEXTANT. 

38. The artificial horizon is any level reflecting surface. Al- 
though the surface of still water, of pure fluid pitch, &c. will 
answer the purpose, mercury is found to be the best. It is usu- 
ally contained in a shallow wooden vessel, and a roof, the two 
sides of which are of glass, protects it from the wind. The in- 
cident ray passes through one glass nearly perpendicularly, and 
the reflected ray passes out at the other. This instrument gene- 
rally accompanies the sextant in astronomical observation. 

39. The floating collimator is a late invention, and consists of 
a small telescope supported on a float in a vessel of mercury. A 
visible point of light may be placed in its focus, the rays diverg- 
ing from which, after passing through the object-glass emerge 
parallel, and it may therefore be viewed as an infinitely distant 
star, by a telescope attached to any mural or other vertical circle. 
Since the axis of the floating telescope always preserves the same 
inclination to the horizon, a reversed observation on opposite 
sides of the fixed circle fixes the zenith point of that circle. 
There are two forms of this instrument, the horizontal and ver- 
tical, terms which designate the position of the floating telescope. 



CHAPTER III 



OF THE SEXTANT ALTITUDE AND AZIMUTH INSTRUMENT EQUA- 
TORIAL. 

40. The sextant. — By adapting the principle of reflection to 
the measurement of angles, a number of instruments have been 
invented, which are independent of any fixed support, and may 
therefore be used in any situation, however unsteady. The sex- 
tant is the most common of these instruments, and its peculiar 
importance to the navigator, no less than its general utility in the 
observatory, is sufficient reason for dwelling at some length on 
its several adjustments, and on the manner of using it in the meas- 
urement of angles. For a description of the principal parts of 



THE SEXTANT. 35 

the instrument the student may refer to Art. 129 of Olmsted's 
Astronomy. 

41. The principal adjustments of the sextant are as follows: 

(1.) To make the index-glass perpendicular to the plane of the 
sextant — Move the arm ID (fig. 15.) from zero near F towards 
the middle of the arc ; and turning the limb or arc FE from the 
eye, look at its reflection in the index-glass I. If the reflected 
portion of the limb is a perfect continuation of the part seen by 
direct vision, the index-glass is truly perpendicular ; but if not, 
it must be made so, by alternately loosening and tightening the 
screws behind it. But this adjustment in a good sextant will 
seldom be found deranged, except by violence. 

(2.) To set the horizon-glass perpendicular to the plane of the 
sextant. — Screw in the telescope T, and point it towards a star. 
Move the index-arm backwards and forwards past the zero of 
the limb, and if the two images of the star do not exactly coin- 
cide in passing one another, turn a screw at the top or bottom 
of the horizon-glass H until such coincidence can be made. 

(3.) To make the horizon-glass parallel to the index-glass when 
the index is at zero. — When of the vernier is at of the limb, the 
horizon and index glasses should be parallel, and the two images 
of a star should perfectly coincide. If they do not, adjust by 
the screw at the side of the horizon-glass. This adjustment 
should be repeated alternately with the last till both are as per- 
fect as may be. 

(4.) To set the axis of the telescope parallel to the plane of the 
sextant. — There are two parallel wires on opposite sides and 
equidistant from the centre of the field of the telescope, and usu- 
ally crossed by two others. Turn either pair around until they 
are parallel to the plane of the instrument. Adjust by the screws 
in the ring which holds the telescope, until the images of two stars 
more than 90° distant from each other, having been brought to 
perfect coincidence on one wire, shall remain so on the other. 

These adjustments may be performed, although less accurately, 
in the day time by means of the sun, and the dark glasses of the 
instrument are then to be employed. Of these there are two 
sets ; one between I and H, and perpendicular to the reflected 
ray IH ; the other just beyond H, in the line TS. As many as 



36 THE SEXTANT. 

are needed of the first set may be turned up when the image of 
the sun is received by reflection, or of the other when it is view- 
ed directly. The two surfaces of each glass in a good sextant 
should always be perfectly plane and parallel, and are proved to 
be so, if the two images of the sun, when brought into exact 
contact, remain so after the glass under trial has been taken out 
and reversed. 

42. On the completion of these adjustments, the sextant be- 
comes an accurate instrument, and may be employed in angular 
measurement. But the observer must first know — how to move 
the index-arm in measuring any distance, — how to read off the 
angle when measured, — and how to determine the index error. 

(1.) To move the index-arm in measuring angles. — There are 
two screws attached to the moving extremity of the index-arm 
ID; one may be seen in fig. 15, and is beneath the limb; the 
other (not represented in the figure) is called the tangent screw, 
and lies in the direction that its name implies. The first fastens 
the arm after it has been shifted nearly to the point desired, and 
the tangent screw, acting only when the other is fixed, serves by 
a slow motion to bring the images into perfect contact. This 
screw moves the arm but for a short arc of the limb, and the 
observer should be particularly careful not to force it at its limit 
of action. 

(2.) To read off the value of a measured arc. — Observe what 
minute and part of a minute on the vernier coincides exactly 
with a division on the limb ; and add this arc to the degree and 
minute on the limb immediately preceding the first division on 
the vernier. (See Chapter II. on " the Vernier.") 

(3.) To find the index error. — The adjustment, No. (3,) will 
render the two images of the same star very nearly coincident 
when the index is at 0. But since they cannot be precisely so, it 
becomes important to know at what point of the arc exact coin- 
cidence takes place, since it is from this point that all angular 
measurement is reckoned as from the true zero. Therefore by 
means of the tangent screw, bring the two images of a star to 
perfect coincidence, (or to that point in passing by each other 
where they should coincide,) and read off the measure, calling it 
+ when forward from zero, and — when backward, or towards 



THE SEXTANT. 37 

F. This is the index error, and is always to be subtracted from 
every other angle read off on the limb, paying attention to its 
sign. 

Another method is to bring the two images of the sun into 
exact contact on one side, and then to make the reflected image 
pass the other and touch on the opposite side. The readings in 
the two cases must be marked + or — , according to their posi- 
tions with regard to zero, and half their algebraic sum will be the 
index error. 

43. To measure the diameter of the sun. — Proceed as in obtain- 
ing the zero error by means of the sun, and half the algebraic 
difference of the two readings will be the sun's diameter. In 
this case, two measures are taken in opposite directions from 
zero, and it is therefore needless to apply the zero error. 

To take an altitude of a star or of the sun by reflection from mer- 
cury. — Set the index near zero, put in the telescope, and make 
the wires parallel to the plane of the instrument. By means of 
the handle, hold the instrument with the right hand, with its face 
to the left, and in the vertical plane of the star, towards which 
let the telescope be pointed. Two images will be seen in the 
field of view, one of which, viz. that formed by reflection, will 
apparently move downward when the index is pushed forward. 
Follow the reflected image as it travels downward, until it ap- 
pears to be as far beneath the horizon as it was at first above, 
and the reflection of the star from the mercury also appears in 
the field of view ; then fasten the index, and make the contact 
perfect by means of the tangent-screw, taking particular care 
that the images shall be midway between the parallel wires. 
The reading of the limb, diminished of course by the index er- 
ror, will be twice the apparent altitude of the star at that moment. 

To manage the sextant dexterously, the observer must acquire 
the art of giving it a swinging motion, as if turning on the axis 
of the telescope, w r hich may be gained most easily by leaning 
the body over gently to the right and left alternately. The image 
reflected from the index-glass may thus be made to sweep the 
arc of a circle, convex downwards, and in slowly passing and re- 
passing the image seen in the mercury, the contact may be very 
accurately judged of. 



38 THE SEXTANT. 

In taking an altitude of the sun, proceed as with a star, turn- 
ing up dark glasses between H and I, and also beyond H, for the 
protection of the eye.* If the altitude of the lower limb is to 
be observed, and an inverting telescope is used, make the image 
reflected from the index-glass sweep under that seen in the mer- 
cury, turning the tangent-screw until it just touches in gliding 
by, and vice versa. The image in the mercury can always be 
distinguished from the ■ other by its remaining fixed, and being 
often affected with tremors. 

The beginner, in attempting to bring down the star to its image 
in the mercury, will probably lose it, and be obliged to commence 
anew. If therefore he has the means of knowing the approxi- 
mate altitude, he may set the index at twice that angle, and point 
the telescope at once to the star in the mercury. The swinging 
motion will then make the reflected image pass horizontally 
through the field. 

To take an altitude by means of the natural horizon. — If the 
observer is at sea, the natural horizon must be employed, and the 
arc measured, (after subtracting the index error, dip, and refrac- 
tion,) will be the altitude. The star or sun's limb must be made 
to graze the horizon by the swinging motion. This method of 
taking altitudes is sufficiently accurate for the navigator, but not 
for the astronomer. 

To find the distance between the moon and sun, or between the 
moon and a star. — The same management is necessary here as in 
taking an altitude, except that instead of holding the sextant in a 
vertical plane, it must be held in the plane passing through the 
two objects and the eye of the observer. When the index has been 
set to the approximate distance, or brought thereto by following 
the reflected image, the swinging motion will bring the sextant ex- 
actly into this plane, and the two bodies will appear together. 
Let the reflected limb of the sun rise and fall by that of the 
moon until perfectly tangent to it, as in observing altitudes. So 
with a star, make the moon's reflected limb just touch it in swing- 
ing by. A little practice, or a rule similar to that prescribed in 
taking altitudes, will enable the observer at once to decide, whe- 

* A combination of a light red and green glass commonly gives an image less un- 
pleasant and heating to the eye, than the darkest image that can be made by red 
glasses alone. 



THE ALTITUDE AND AZIMUTH INSTRUMENT. 39 

ther it is the distance between the nearer or farther limbs of the 
moon and sun that has been taken. 

The use of the sextant in connexion with the artificial horizon, 
will be again alluded to in the chapter on " the methods of deter- 
mining the latitudes and longitudes of places." 

Of the Altitude and Azimuth Instrument. 

M. Instruments of this kind, however varied in their form, 
consist (1.) of a graduated circle confined to a horizontal plane, 
and turning freely in that plane along with its vertical axis ; (2.) 
of another graduated circle secondary to the former, attached to 
its vertical axis, and capable of being brought into any vertical 
plane by its motion ; and (3.) of a telescope firmly fastened to the 
secondary circle in its own plane, and turning with it in altitude. 
Each circle is read off by verniers or reading microscopes, usu- 
ally three in number, and the mean result of such readings is re- 
corded after each observation. 

By turning the instrument so that the intersection of its cross- 
wires shall exactly coincide with the image of a star, and noting 
the instant of such bisection, the altitude and azimuth of the star 
at that moment will be obtained from the readings of the circles. 
But it is first necessary to determine the points on the circles 
from which the reckoning commences. The meridional point 
on the azimuth circle is its reading when the telescope is pointed 
north or south, and may be determined by observing a star at 
equal altitudes east and west of the meridian, and finding a point 
half way between the azimuthal angles recorded in the two cases. 
The horizontal point of the altitude circle is its reading when 
the axis of the telescope is horizontal, and may be found by aid 
of the level, plumb-line or collimator, or by alternate observation 
of a star directly and reflected from mercury, taking a mean be- 
tween the two recorded angles. 



■&■ 



45. This instrument, in its capability of determining the place 
of any star above the horizon at any hour of the night, has an 
advantage over such as are confined to the meridian, like the 
transit instrument and mural circle. But stars are not located 
in the heavens by altitudes and azimuths, because these are con- 
stantly changing from moment to moment ; and measures with 



40 THE EQUATORIAL. 

this instrument are therefore not complete without a record of 
the instant of observation, and must be reduced to right ascen- 
sions and declinations by processes of spherical trigonometry, 
laborious when accumulated. This circumstance tends to con- 
fine the use of the instrument in practice to meridional observa- 
tions ; and here its construction gives it a great advantage in 
taking repeated measures of right ascension and declination a 
little on each side of the meridian, which admit of reduction to 
that circle by easy formulae, and have all the weight of accumu- 
lated observations. 

Of the Equatorial Instrument. 

46. If an altitude and azimuth instrument be turned from its 
upright position so that its axis, instead of pointing to the zenith, 
shall be directed to the pole of the heavens, it becomes an equa- 
torial. What was before the azimuth circle, now lies perma- 
nently in the plane of the equator, and the altitude circle in the 
new arrangement can be turned into no position, in which it will 
not coincide with a horary circle, or one of declination. The 
circles are called respectively the horary and declination circles ; 
and the former is usually graduated into hours, minutes and 
seconds of time, instead of degrees, minutes and seconds of space, 
like the azimuth circle. 

To determine if an equatorial is in approximate adjustment, — 
follow a star in its diurnal course by means of the horary circle, 
the declination circle remaining clamped. The star ought al- 
ways to pass the field of view at the intersection of the cross- 
wires at all hour angles, or rather, to pass apparently below them 
by the amount of vertical refraction due to its altitude. 

To find a star, — set the telescope to the declination of the star 
in the meridian position, and then turn the hour circle to the 
star's horary angle at the moment, which is always the time 
elapsed since it last passed the meridian, and is equal to the side- 
real time then shown by an adjusted clock, minus the right as- 
cension of the star. It is best in practice to set the hour circle 
at a horary angle corresponding to an instant two or three mi- 
nutes in advance, to allow of time for preparation. 

To determine the approximate right ascension and declination 
of an unknown object, — bring the object on the intersection of 



THE EQUATORIAL. 41 

the cross- wires, and note the corresponding instant of time shown 
by the clock ; the reading of the declination circle will be its de- 
clination, and its right ascension, (if the hour circle is graduated 
from west to east up to XXIV h uninterruptedly,) will be the cor- 
rected sidereal time of observation, minus the reading of the 
hour circle. The result will not be very accurate unless the in- 
strument is in good adjustment, the errors of the zero points of 
both circles allowed for, and the corrections for refraction, espe- 
cially at low altitudes, applied. 

47. The oblique position of the circles of this instrument and 
of their axes, renders it nearly impossible so to combine mate- 
rials, as to prevent unequal strain and bending of its parts. This 
defect, though very minute in amount, is sufficient to exclude the 
equatorial from the highest accuracy of observation, and refer 
the formation of stellar catalogues to instruments more symme- 
trical in their positions relative to the horizon, — a circumstance 
much to be regretted, since no instrument, except the transit- 
circle, could furnish the requisite data more easily and rapidly. 
The equatorial, however, is nevertheless exceedingly useful in 
determining small differences of right ascension and declination, 
such as between objects in the same field of view ; — in enabling 
the observer to follow with ease and measure with certainty 
double stars and other objects, and to settle their places at the 
same time so as to be recognisable thereafter ; — and in fixing 
with sufficient accuracy the places of comets and other bodies, 
which cross the meridian at a time when it is either impossible 
or inconvenient to observe them. For its peculiar adaptation to 
these important uses, the equatorial is in very frequent demand, 
and holds a high place among astronomical instruments. 

6 



42 THE TRANSIT INSTRUMENT. 



CHAPTER IV. 

OF THE TRANSIT INSTRUMENT. 

48. This most important of astronomical instruments, in con- 
nexion with a clock or chronometer, is applicable to such a va- 
riety of purposes, and especially gives the observer so complete 
a command over that primary element, the time, that we may be 
permitted to consider at some length its several adjustments, the 
means of rectifying them, the conduct of actual observation, and 
the best mode of eliminating instrumental errors. In the body 
of the chapter, such minuteness is not aimed at, as would be un- 
interesting to the more numerous class of students, who are de- 
sirous of understanding only the general method of proceeding 
in an observatory, or such refinement, as would discourage 
any one, unaccustomed to mathematical details ; while a few 
pages are appended, by which the observer will be enabled to 
choose his mode of proceeding, and reduce his observations, so 
as to obtain the most accurate results within the reach of his in- 
strument. 

49. Location of the instrument. 

Let us suppose that the reader is in possession of a good port- 
able transit and clock, and is desirous of using them to advan- 
tage. The situation most suitable for the instrument would en- 
gage his earliest attention. And first, it should have a firm basis ; 
the pier on which it stands should be of stone, brick, or othei 
solid material, — should descend into the ground several feet, so 
as not to be affected by frost, or by tremors arising from the 
vicinity of public roads, — and should rise only so high above the 
ground, as to afford a clear view of the north and south points. 
The next desirable requisite is, that it should command a view 
of the whole, or a considerable portion of the meridian. The 
room, therefore, in which it stands should have openings in the 
roof in the line of the meridian ; these should be about 12 or 15 
inches wide, and should be continued down the north and south 
walls so as to afford a clear sweep from the north to the south 



THE TRANSIT INSTRUMENT. 43 

point of the horizon. They should be closed by doors easily 
opened from below, and should be secure against the admission 
of rain or snow. If such a situation as this cannot be procured, 
the transit may be placed on a solid pier out of doors, and guard- 
ed by a tight cover from the weather ; or if it can be firmly 
placed in a north or south window, so as to command a range 
of 60° or more in altitude, it may be used so as to afford very 
accurate results. A south window is preferable to a north, espe- 
cially is it commands the passages of the sun, moon, and planets. 

50. Description of the mechanical contrivances for adjustment. 

The general construction of the frame of the instrument may 
easily be understood by an inspection of fig. 16, or by reference 
to Olmsted's Astronomy, Art. 121. The two ends of the axis 
are called pivots, and great care is taken to make them exactly 
equal and cylindrical ; they rest in angles of polished steel, or 
other very hard substance, shaped as at Y, (fig. 14.) because con- 
tact at only two points in the circumference of the pivot ensures 
greater accuracy of meridional motion. These angles are called 
Y's, from their resemblance in shape to that letter. At one end 
of the axis there is usually a screw, by which the Y of that ex- 
tremity may be raised or lowered a little, and thus the axis be 
made perfectly horizontal. This adjustment is made by means 
of the level, (Chap. II., Art. 36,) and is called the adjustment for 
horizontality of the axis. Another screw moves the Y at the 
other extremity backwards or forwards, and by its azimuthal 
motion is of use in bringing the telescope into the plane of the 
meridian, when a little east or west of it. This adjustment is 
termed that of position in the meridian. When the frame of 
the instrument rests upon three feet screws, one of them may 
supply the place of the screw of elevation. The perpendicular 
wires of the telescope are either three, five, or seven in number, 
but most commonly five ; in small instruments only one horizon- 
tal wire is inserted. To make these wires exactly perpendicular, 
there is a contrivance for moving them around with a circular 
motion in their own plane ; and by another arrangement they 
admit of lateral motion for the adjustment of collimation. These 
contrivances are different in different instruments. The field of 
view is illuminated at night by a lamp placed so as to shine into 



44 THE TRANSIT INSTRUMENT. 

one of the pivots, the axis being perforated for this purpose. 
An oval ring of painted metal, or card paper, placed at an angle 
of 45° within the junction of the axis and telescope, and encir- 
cling without obstructing the pencil of light from the object- 
glass, turns the light down the tube, and makes the wires appear 
black on a bright ground, yet does not obliterate the brighter 
stars. 

The axis is furnished at one of its extremities with a circle 
and vernier, by which the telescope can be directed to any re- 
quired altitude or declination. The vernier usually admits of 
being unloosed and refastened so as to point to different portions 
of the circle while the telescope remains stationary. 

All the screws of adjustment, and indeed, all parts of the tran- 
sit, should be fitted without shake or improper motion ; and when 
the adjustments are once made, they should be as permanent as 
possible. For this purpose, clamps or tightening screws are at- 
tached to such of them as require it, and these should always be 
attended to after adjustment. 

With this brief description of the more important parts of the 
instrument, we will now proceed to the adjustments, which should 
succeed each other in the following order : 

51. Distinctness of vision and parallax. 

The optical adjustments of the telescope are the first to be 
examined. The system of wires or spider-lines is in a plane per- 
pendicular to the axis of the tube, and set in a circular aperture 
very near the eye-end. It should be in the common focus of the 
eye-glass and object-glass. First, to place the lines in the focus of 
the eye-glass, push in or draw out the eye-tube until they are seen 
with perfect distinctness. Next, for the purpose of throwing the 
focal image of the object-glass in the exact plane of the wires, ei- 
ther the object-glass or the wires are set in a tube which admits 
of motion within the main tube,. and which is fastened securely 
by a fixing screw after the proper adjustment has been obtained. 
This is only the case when (the wires appearing perfectly well 
denned by the first adjustment) the images of objects at the dis- 
tance of one mile or several are also as distinct as possible. A 
still surer test of the perfection of this adjustment arises from 
the circumstance, that if the image be cast either before or be- 



THE TRANSIT INSTRUMENT. 45 

hind the lines, a parallax will be detected by moving the eye 
laterally, and distant objects will move on the lines. The images 
of near objects, however, ought neither to be entirely distinct 
or motionless, when the telescope is in perfect order. These ad- 
justments are called the adjustments for distinctness of vision and 
parallax. 

The optical excellence of the telescope may now be tested as 
directed in the chapter on " the Telescope," Art. 8. Any im- 
perfections in the object-glass or eye-piece, which may render 
either the star or wires distorted or ill denned, especially if they 
exhibit a star of the first magnitude otherwise than round, will 
tend to diminish the accuracy of observation. After the transit 
has been placed in the meridian, and the wires adjusted as de- 
scribed hereafter, let a star run occasionally upon the horizontal 
wire, and if it does not remain perfectly bisected while the eye 
is moved up and down, the adjustment for parallax is not quite 
perfect. 

52. Horizontality of the axis. 

In our illustration of the use of the level, (Chap. II., Art. 36,) 
we have given ample directions for levelling the axis of the transit. 

We have remarked that in a perfect instrument, the pivots are 
exactly equal and round. The level is competent to detect any 
imperfection in these respects. For suppose we take the transit 
axis from its Y's, and reverse it end for end ; if the bubble does 
not give the same indications in the two positions of the axis, but 
shows a tendency towards either pivot, that pivot is of larger 
diameter than the other. And if the level be made to bear 
against a fixed support while the telescope is turned in altitude, 
any motion in the bubble will prove that one or both of the pivots 
are not exactly round. These are not accidental errors, but in- 
herent faults of the instrument, and recourse to the original ma- 
ker or to an excellent workman is the only means of remedy. 

The level is liable to give erroneous indications, unless it hangs 
always with the same face upwards. If, therefore, by swinging 
it gently back and forth, the position of the bubble is materially 
altered, a cross level should be firmly attached to it, by means 
of which the same curve of the tube may always be made up- 
permost. 



46 THE TRANSIT INSTRUMENT. 

53. Perpendicularity of wires. 

Some well defined terrestrial point may now be brought upon 
the intersection of the wires in the centre of the field by the screw 
of motion in azimuth. If, on turning the telescope in altitude, 
this point is perfectly bisected by the central wire from top to 
bottom, the wire is perpendicular to the horizontal axis. If not, 
the ring or tube containing the wires must be turned in a circu- 
lar direction until it is so bisected, and there fastened. 

54. Collimation in azimuth. 

That point of an object-glass, through which a ray in passing 
suffers no refraction, is called its optical centre. A line drawn 
from this point to the central vertical spider's line is called the 
line of collimation in azimuth, and ought to revolve in the plane 
of the meridian. But if it is inclined to the transit axis, it will 
not trace out any great circle, but some small circle of the hea- 
vens. Suppose that the line of collimation produced to meet the 
same terrestrial point as before, leans towards the eastern pivot. 
By reversing the transit, it is evidently made to lean as far to the 
west, as it did before to the east, and the central wire will be 
thrown entirely off of the point which it before bisected. The 
true direction of a perpendicular to the axis is obviously half- 
way between its present and its former position. Therefore, by 
the contrivance for the lateral motion of the wires, make the 
central wire traverse half the distance by estimation from its 
present position to the point it first covered. To correct remain- 
ing error, bring the central wire again upon the distant point of 
reference by means of the screw of motion in azimuth, and re- 
peat the process, until it is bisected from top to bottom of the 
central wire equally before and after reversion. 

55. Collimation in altitude. 

If the circle attached to the instrument is intended to indicate 
meridian altitudes, take the declination of any bright star that 
crosses south of the zenith during the evening, from the Nautical 
or American Almanac, calling it + if north, and — if south of the 
equator. Add to it the elevation of the equator to obtain its true 
altitude, and then the refraction due to its altitude. The sum will 
be the apparent meridian altitude. While the star is running 



THE TRANSIT INSTRUMENT. 47 

along the horizontal wire of the transit instrument, (supposed to 
be in the meridian, or very nearly so,) unloose the vernier, set it 
to the apparent altitude, fasten it firmly, and then see if the star 
is still on the horizontal wire. If declinations are indicated by 
the circle, the vernier must be set to the true declination, increas- 
ed by the same refraction as before. By this process, the ver- 
nier will probably be correct within 2' or 3', and its zero error 
may be determined and allowed for. If the vernier be now set 
to the apparent altitude or declination of any expected star, the 
star will enter the field on or near the horizontal wire. 

56. Position in the meridian. 

This is the most difficult of the adjustments of the instrument, 
and requires that all the others should be first completed. In ex- 
plaining the methods of adjustment, the clock will be considered 
as indicating sidereal time, in which case the right ascension of 
each star, as it arrives at the meridian, will be the same as the 
clock time at that moment. If, therefore, the pendulum of the 
clock has not been adjusted to the proper length, it should be 
shortened or lengthened until a star comes to the intersection 
at the centre of the field within a few seconds of the same time 
on two successive evenings. This may be done before the tran- 
sit is brought very near a meridional position. 

By the pole star. — When the telescope commands the northern 
portion of the meridian, this is the easiest and best mode of ad- 
justment. First point it to the pole star, and then turn it to some 
other star about to cross the meridian at a distance from the pole. 
At the moment of its central passage, set the clock to its right 
ascension, and it will thenceforth indicate nearly sidereal time. 
The approximate times of the upper and lower culmination of 
the pole star are then known, being the clock times answering to 
its right ascension, and 12 hours thereafter. For a few minutes 
before and after either of these moments, on account of its ex- 
tremely slow motion, it is almost exactly in the meridian. Fol- 
low the pole star therefore by turning the transit, till it arrives 
within half an hour or less of the meridian. The base of the 
frame may then be fixed by wedging or pouring hard cement 
underneath, if not sufficiently steady, and the horizontality of 
the axis should be tested. Still follow the star by means of the 



48 THE TRANSIT INSTRUMENT. 

screw of motion in azimuth until the clock shows its R.A., or 
its R. A. + 1 2 h . The central wire then, if the previous adjustments 
have been well attended to, will almost precisely coincide with 
the meridian of the place, even if the clock be 2 m or 3 m in error. 
Now, the axis being perfectly horizontal, and the line of col- 
limation perpendicular to it, if the central wire by its motion 
bisect the small circle described by the pole star, the adjust- 
ment is complete. This will be the case, when the interval be- 
tween the upper and lower is equal to that between the lower 

24 h 
and upper transits, each being = — -. Few portable instruments, 

however, are competent to render this star visible in the day 
time. Hence it is usually best in practice to rectify the clock 
or ascertain its correction the next evening, by stars at a distance 
from the pole, and repeat the process with a more perfect know- 
ledge of the sidereal time. 

By a pair of circumpolar stars. — Choose two stars which cross 
the meridian within a few minutes of each other, one above, and 
the other below the pole. Let A and T be the R. A. and ob- 
served time of passage of the upper star, and a and r those of 
the lower. Then when the central wire of the transit coincides 
with the meridian, A— a=T— <r-fl2 h ; when the wire deviates 
to the west at the north of the zenith, (A— a)<(T— r+12 h ), and 
vice versa. This will enable the observer* to determine in which 
direction he must move the screw for azimuthal motion. Sup- 
pose he finds (A — a)>(T— r-f 12 h ) by 5 m , and in consequence 
thereof moves the telescope westward by 4 turns of the screw. 
If now on the following evening, he finds that upon the same pair 
of stars (A— a)>(T— r+12 h ) by l m , since the excess has been di- 
minished 4 m by 4 revolutions of the screw, it is evident that 
one more revolution in the same direction will place it very 
nearly in the meridian. 

By a high and low star. — This method is peculiarly adapted to 
instruments which have only a south exposure. Choose two stars, 
one above, the other below the equator, and differing 40° or more 
in declination, and a few minutes or less than an hour in right 
ascension. Let A and T, a and r, represent the same quantities 
as before for the northern and southern stars respectively. Sup- 
pose the plane of the telescope south of the zenith deviates to 



THE TRANSIT INSTRUMENT. 40 

the east, but still by the application of the level is made a verti- 
cal circle, and of course, cuts the meridian at the zenith, and de- 
parts more and more from it towards the south. The southern 
or lower star obviously must be longer in crossing from the tran- 
sit plane to the true meridian, than the upper star, which crosses 
where the interval is narrower : the time of its passage is there- 
fore earlier than it should be in comparison with the other ; that 
j Sj T— t is greater than it should be, or A— a<T— r, when the 
deviation is eastward, and vice versa. The deviation should be 
corrected in the same way as by the last method. It may be 
proper to observe, that in this, as well as in the two preceding 
methods, the best way of correcting the error is by ascertaining 
its exact amount, which is easily obtained from the observed 
passages of the stars by the formulae near the end of the chapter.* 

57. Location of the meridian mark. 

When the instrument has been once fairly brought to the me- 
ridian, a mark may be placed either to the north or south, or 
both, for the advantage of constant and speedy reference. It 
should be placed at such a distance, as not to be affected by par- 
allax, (Art. 51,) and yet not so far as to be imperfectly seen. An 



* For, by form. 1, Art. 72, we have for the two stars, 

a =-.t-\-x-\-a sin (<p — h) sec S for one star, 
and a'=t'-\-x-\-a sin (<j> — 6') sec 5' foi the other, 
where the two other errors b and c are supposed to be nothing. 

Subtracting, we have a— a'=t— i'-\-a j sin (<p — 6) sec 6— sin (<£ — <3') sec 6'\ 

and a= — : — ■ or the azimuthal deviation. 

sin (<p — 5) sec 6 — sin (0 — <5') sec <5' 

For instance, take the stars a Piscis Australis and a Pegasi, an excellent pair of 
high and low stars, differing in right ascension by 8 m . By our Table V. for New Ha- 
ven, sin (<t> — 5) sec S equals 

for a Pise. Aust. 1.102 

for a Pegasi, 0.468 



0.634 

which is therefore a constant divisor for these two stars at that place. Thus if the 

observed times of their passage, Jan. 1, 1840, differ by -|-8 m 30 8 .5, then (a — a') — 

(t—f) =-}-7 m 59«.8-8 m 30". 5=-30 8 .7=0.634a, and a, or the azimuthal deviation, 

— 30 8 .7 
equals , = — 48".4. Therefore if the equatorial interval between each two of 

the wires is 64', the telescope must be pointed to a terrestrial object on the southern 
horizon, and screwed eastward about £ of one of the intervals. 

7 



50 THE TRANSIT INSTRUMENT. 

excellent form of the meridian mark is a piece of sheet iron or 
copper 6 or 8 inches in diameter, cut into the shape of a hol- 
low equilateral triangle ; this, when placed on an eminence, so 
as to have the sky in the background, may be very perfectly bi- 
sected at the vertical angle by the central wire of the instru- 
ment. It should admit of being moved laterally, if not at first 
exactly placed in the meridian. 

58. The clock, — its rate and error. 

The clock is the indispensable companion of the transit instru- 
ment. The office of the transit is in fact, to point out the agree- 
ment or disagreement between the great and perfectly regular 
time-piece furnished by the apparent revolution of the sphere, 
and the irregular and imperfect clock of the observatory. It is 
available only as a plane of reference, an indicator or pointer to 
the grand motion of the sidereal heavens, thus rendering it com- 
parable with the measurement of time by mechanism of human 
construction. 

No clock is fit for the nicer purposes of astronomy, unless it 
is rendered as invariable as possible. For this purpose, much 
attention is always devoted to the construction of the escape- 
ment, which should not be such, as to allow of any inequality in 
the transmission of the moving power to the pendulum. Still 
more should the pendulum be of that class, called compensated ; 
and so be free from the variable effects of heat and moisture. 
The two forms usually adopted for astronomical uses are what 
are commonly termed the gridiron, and the mercurial, (Art. 365, 
Nat. Phil.) Yet, if no other is attainable, a common clock with 
increased attention and labor may under favorable circumstances 
afford valuable results. In the use of the clock, especially if un- 
compensated, it is a principle of cardinal importance, to make 
the intervals of time which depend upon its regularity as short 
as may be, as for instance, that between a pair of circumpolar, 
or high and low stars, (Art. 56.) And in observing any pheno- 
menon of importance, as an eclipse, occultation, or transit of the 
moon's limb, the clock should be compared with the heavens by 
means of multiplied transits of stars at no long intervals both 
before and after the event. The heavens are always to be con- 
sidered as a time-piece of perfect uniformity, with which it is 



THE TRANSIT INSTRUMENT. 51 

the interest of the observer carefully and closely to compare his 
own whenever he is forced to depend upon the latter. 

59. If the clock is regulated to sidereal time, it is supposed in 
theory to keep exact pace with the stars, and to indicate at any 
moment the right ascension of the star then crossing the meri- 
dian. But in practice, it has both an error and rate. An ex- 
ample will best explain the meaning of these terms. If a Arietis,* 
in R.A. 2 h m , cross the meridian at 2 h m 30 s by the clock, the 
clock is fast of the heavens 30 s , or its error =+30 s . The correc- 
tion of the clock, or the quantity to be applied to its indications 
to obtain the right ascension of the star, is the same in amount, 
but with an opposite sign, — viz. in this case = — 30 s . Now if 
another star a Aurigae in R.A. 5 h m , cross 3 h after at 5 h m 33 s , 
and if the time of passage of 15 Hydrae, in R.A. 8 h m , on the 
same evening be 8 h m 36 s , it is very plain that the clock's error 
is increasing, or in other words, its rate =+3 s in 3 h , and +6 s in 6 h , 
or +24" per day. The error of the clock then, is its difference 
from true sidereal time at any given moment, + if faster, — if 
slower ; its rate is its daily gain or loss on sidereal time, -f if 
the former, and — if the latter. If the error at a given time be 
called e, and the daily rate r, the error at any time thereafter 
will of course be e-\-tr, where t denotes the number of days and 
parts of a day elapsed since e was determined. All that is neces- 
sary to find the true time at which any event happened, is to ap- 
ply the correction of the clock, or — e— tr to the observed time. 

60. The rate and error are not by any means due to imper- 
fection of the clock, and may be as large or larger in the most 
perfect as in the rudest time-piece. Yet it is convenient in prac- 
tice that they should both be of small amount. The error may 
be nearly annihilated by setting the clock to the right ascension 
of a star, and starting the pendulum at the moment when that 
star is on the middle transit wire ; and the rate may be readily 
brought within small limits, as l 8 or 2 s per day, by lowering the 
bob of the pendulum when positive, and raising it when nega- 
tive. The number of turns by which the screw of the pendu- 

* These stars have nearly, but not exactly, the right ascensions which are here as- 
signed to them in round numbers for the sake of illustration. 



52 THE TRANSIT INSTRUMENT. 

lum is altered, may be compared with the consequent alteration 
of rate, and thus the ultimate reduction very certainly effected. 
After the rate has been reduced to a small quantity, say less than 
I s , it is better to let the error accumulate than to stop the clock. 
Indeed, it ought never to be stopped or to be suffered to run 
down, when in frequent use. 

Whatever may be the error and rate, the clock is perfect if 
the rate is uniform, or equal in equal times. But in common 
clocks, the rate may be nothing for one day, and several seconds 
+ or — , the next ; and in ordinary uncompensated watches, this 
difference on different days may amount to 2 or 3 minutes. The 
latter cannot therefore be trusted for an accurate knowledge of 
the time. But with a compensated clock of the first order, an 
alteration of a second in the rate during a single day is scarcely 
to be apprehended. 

61. Method of observing and registering transits. 

We have hitherto supposed our observer to be so far acquaint- 
ed with some method of observation, as to be able to note the 
times at which stars in the field cross the middle wire. A more 
particular explanation of the best modes of conducting observa- 
tion becomes now necessary. 

The field of view should be illuminated by the lamp mention- 
ed in Art. 50, until the wires are perfectly and sharply visible. 
The vernier should be set so as to indicate the place of an ex- 
pected star on the circle, allowing for refraction, (Art. 55,) and 
it will then enter the field at one side nearly upon the horizontal 
wire. In northern latitudes, the star enters upon the right hand 
side of the field if between the zenith and southern, or between 
the pole and northern horizon, — but if between the pole and 
zenith, it first appears on the left hand edge, and departs at the 
right ; its apparent line of motion being inverted by the tele- 
scope. The stars should always be made to cross the same points 
on the perpendicular wires, that minute errors arising from their 
want of parallelism to each other and to the meridian may be 
avoided. 

62. The star on entering the field will move slowly across it 
in a horizontal direction, and it is the business of the observer to 



THE TRANSIT INSTRUMENT. 53 

note the times of its passage across the transit wires with the 
utmost accuracy. The clock should be so placed, and its face so 
well illuminated, that the observer, stationed at the transit, can 
at any moment read the second indicated by the pointer. The 
second last counted before the star crosses the wire, is to be re- 
gistered as the even second of passage; and the fractional part 
of the next second which elapses before the instantaneous bisec- 
tion of the star may be most easily judged of by the eye, in com- 
paring the small intervals of space by which the star is short of 
and beyond the wire, at the instants of the preceding and follow- 
ing beats. Thus, suppose the observer takes up the second 2 
from the clock, and goes on silently counting 3, — 4, — 5, — 6, &c, 
in exact coincidence with the beats as he turns to the field of 
view ; if the star appears at the points 6, 7, 8, (fig. 17,) as he hears 
the corresponding seconds from the clock, 7 will plainly be the 
even second to be recorded. And if at the wire A the distance 
7.. .A appears about T V of 7.. .8, as nearly as can be estimated 
when the object is in motion, 7 8 .7 is to be recorded on the journal. 
The minute during which the passage occurred should then be pre- 
fixed, taking care to make such allowance for the time occupied in 
estimating and writing down the seconds and tenths, that there 
shall be no danger in making the record one minute in advance. 

63. The observer will not be long in perceiving that the spaces 
of time occupied by the stars in traversing the intervals between 
the wires, are very different on different points of the meridian ; 
being shortest at the equator, and longer and longer as the star 
is more distant from that circle. The time occupied by a star 
at the equator in passing between any two of the wires is called 
their equatorial interval ; and this time, converted into minutes 
and seconds of space, is the constant arc of a great circle in- 
cluded between such two wires. But a star whose declination is 
+30° or — 30° moves more slowly than one at the equator ; and 
the time in which it passes from one wire to another is equal to 
the equatorial interval, multiplied by the secant of the declina- 
tion.* Consequently, to find the equatorial interval of a tran- 

* Let PP' (fig. 18.) be the axis of the heavens, EQ the equator, and DF a parallel 
of declination ; let PEP' be the meridian of the place, and PIP' one a little inclined 
to it. A star at D moves over the arc DC in the same time that one at E moves 



54 THE TRANSIT INSTRUMENT. 

sit ; — multiply the interval found from the passage of any known 
star by the cosine of its declination. 

For the pole star, and others within 10° or 12° of the pole, a 
modification of this rule becomes necessary ; for these do not 
pass perpendicularly from wire to wire, but describe consider- 
able arcs of their respective small circles, as in fig. 19. Suppose 
the pole star describes the arc AS in 8 m ; the arc AS of course 
=2° of a circle whose radius is cos 8, and AC= e = sin 2° cos <5, 
since AC is the sine, or very nearly the sine, of the arc AS. 

64. In registering the times of observed transits, no rule is 
necessary other than to have a proper regard for convenience 
of reduction. The form of registry generally adopted by as- 
tronomers, is exhibited in our example, (Art. 76.) The best mode 
of obtaining the mean of the wires when 5 are used, is by the 
following rule : 

Add together the seconds of the Jive transits, and multiply by 
the decimal .2, adding or subtracting from the product as many 

(60 s \ 
= -— 1, as will render it nearly the same as the number 

of seconds at the middle wire. 

over EI ; but since angle EOI = angle DXC .*. arc EI : arc DC : : circ EIQ : circ 
DCF : : EO : DX : : rad : sin PD or cos 6 ; 8 being the declination of the star. 
Therefore DC=EI cos &. 

Now the time in which the star D would traverse the constant space DR or EI be- 
tween the wires must be to the time in which E moves over the same, in the inverse 
ratio of their motions, or as EI : DC : : 1 : cos 8 : : sec 8:1. If then e represent the 
equatorial interval of the wires, e sec 8 will be the interval for a star whose declination 
is 8 ; the wires of the transit manifestly intercepting between them a larger arc of 
the small diurnal circles of stars near the pole, than of the equator. 

Cor. 1. Universally, the time occupied by a star in describing any small space in 
the heavens is inversely as its rate of motion ; that is, is equal to the space X sec 8. 

Thus a star whose declination is 30°, crosses an arc of 6 s in the time 6 s X sec 30°. 

Cor. 2. If two great circles, slightly inclined to one another, cut each other in the 
poles of another great circle, the distance between the two first at any point of their 
circumference, is equal to the arc of the other intercepted between them, multiplied by 
the cosine of the distance of that point from the latter circle. 

Thus, if the circles PEP', PIP , are inclined to one another by the small arc EI, 
the shortest distance between them at D is very nearly equal to EI X cos DE, or EIX 
sin PD. 

This latter corollary is of much use in understanding what effect a small inclina- 
tion of the plane of the transit instrument to the meridian, has upon the time of a 
star's passage. 



THE TRANSIT INSTRUMENT. 



55 



The rationale of the above rule is very apparent ; multiplying 
by 0.2 is equivalent to dividing by 5, and the simple device of 
adding or subtracting multiples of 12 renders unnecessary the 
more tedious operation of adding the minutes as well as seconds, 
and dividing their sum as in compound division. 

When 3 wires are used, divide the sum of the seconds by 3, 

60 s 
and add or subtract — - =20 s , as often as required. 

o 

In transits of the sun, the passages of each limb are to be taken 
and cast up by the above rule as those of two separate objects, 
and the mean of the results will give the passage of his centre. 
But if only one limb is observed, the passage of the centre may 
be inferred by adding or subtracting " the Sid. Time of Semi- 
diameter passing the Meridian," as given on p. I. of each month 
in the Naut. Aim. 

In transits of Jupiter and Saturn, when both limbs are taken, 
the appulses of the first limb may be noted at wires I, III, and 
V, and of the second limb at wires II and IV. If one limb 
only of a planet is observed, the ephemeris must be consulted 
for the time of passage of its semidiameter. 



65. After obtaining from the mean of a great number of tran- 
sits (from those of Polaris especially, if practicable,) the equa- 
torial interval between each two wires respectively, (Art. 63,) if 
these are found to be unequal, a correction is necessary. Thus 
in the instrument which furnished the observations in Art. 76, 
the equatorial intervals deduced from a great number of stars, 
and agreeing also very well with the mean result for that even- 
ing, are as in the first of the following columns : 



Wire 





Eq. Intervals. 


<u 


I— II. 


. . . 63 8 .90 


-C o 


II— III. 


. . . 64 .35 




Ill— IV. 


...64.19 


0} tS 


IV— V. 


. . . 64 .32 


5* 



From mid. wire. From mean ot wires. 



I... 


— 128 s .25... 


. — 128 s .27 


II... 


— 64 .35 . . . 


. — 64 .37 


III. . . 


.00 . . . 


. — .02 


IV. . . . 


+ 64.19... 


. + 64.17 


v.... 


+ 128.51 ... 


. +128.49 



The mean of the transits over the 5 wires is the transit over 
an imaginary line nearly coinciding with the middle wire. In 
the instance before us, this line diners S .02 from the middle wire, 
and is towards wire V. Although the difference is accidentally 



56 THE TRANSIT INSTRUMENT. 

very small in the present instance, and might be safely neglected, 
we shall proceed as we would if it were of large amount. By 
addition of the equatorial intervals we obtain the numbers in 
the 2nd column, the sign — being applied to distances measured 
in a direction opposite to that of the star's motion. One fifth 
of the numbers in the 2nd column =+ S .02, is the distance of 
the imaginary mean line from the middle wire, and applying it 
with changed sign to the distances in the 2nd column, we obtain 
those in the 3rd. 

Now if through inadvertence, or unfavorable weather, the 
transits over only a portion of the wires are observed, the reduc- 
tion to the imaginary centre may be performed by the following 
rule : 

To the sum of the times of transit over the observed wires, add 
the sum of the distances of the unobserved wires from the mean or 
imaginary line,* multiplied by sec 8 ; the amount divided by the 
number of wires observed will be the time of transit over the mean 
line. 

Thus, if a star is observed on wires I, II, and IV only, to 
the sum of the observed times add (— S .02+128 8 .49) sec 8 = + 
128 s .47 sec 8, and divide by 3. 

If II, III, and IV are observed, to the sum of the times add 
(— 128 s .27+128 8 .49) sec 8 = +0 3 .22 sec 8, dividing by 3. 

If the transit over the central wire only be noted, subtract 
(see Note) — S .02 sec 8 from the time of transit. The result 
will be the same as if (—128.27—64.37+64.17+128.49) sec fo» 
+-0 S .02 sec 8 had been added, dividing the result by one. 

This rule will apply to all cases that can be expected to occur. 
Under the pole, it must be recollected, the stars move across the 
wires in an opposite direction from the usual one, and all correc- 
tions to the middle wire must be with changed signs. The same 
must be done, if the transit is reversed, since then the wires are 
also reversed, and are crossed by the star in a different direction 
from before. For example, the distance of wire I from the mean 
is in the present case, 

* These are taken in the present instance from the 3rd column given above. In- 
stead of adding the distances of the unobserved wires, those of the observed wires 
may be subtracted, since the sum of the five must always =0. This course will be 
preferable where but one or two wires are observed. 



THE TRANSIT INSTRUMENT. 



57 



Axis as usual 



above the pole 
below the pole 
above the pole 
below the pole 



— 128 s .27 
+ 128.27 
+ 128.27 

— 128.27 



Axis reversed 

and so for the others. 

In filling up omitted wires of the planets, their motions in the 
intervals must be allowed for; in the 1 case of the moon, this al- 
lowance increases the interval more than g^ti* part, and her par- 
allax at the side wires is also to be taken into account. 



66. A few illustrations taken, with the exception of Polaris, 
from the general example at the end of the chapter, will render 
the application of these rules easy to the learner. 



/ Pegasi. 

33 s .9* 

43.5 

21 h 21 m 53.1 

2.8 

12.5 

145.8 
.2 

29.16 
Add 24 



y Pegasi. 
56 s .8 V. 
3.4 IV. 
h 4 m 9 .5 III. 



69.7 
+198 .80=+192.64 sec 14° 18'. 



3) 268 .50 

89.50 
Subtract 80 



4x20 



21 21 53.16 

Polaris. 
Wire V +128 s .49=+32' 7".4. 



log. sin 32' 7".4 
log. sec 88° 27' 25" 



4 9.50 



0, . . +88° 27' 25". 
+7.9705 
+11.5698 


23 h 40 ra 36 s .5 I. 

21 36.0 II. 

1 1 41 .5 III. 
1 41 35.5 IV. 


+9.5403 

+20° 18' 20" = 


98 45 

+ 81 


29.5 
13.33 


4) 


100 6 


42.83 



1 1 40.71 



* It will probably be found most convenient in practice to set down the transits in 
this columnar form, at the time of observation, in any small note-book the observer 
has at hand. They can then be transferred to the regular journal as exemplified in 
Art. 76, the mean of the wires being previously cast up from the rough columns of 
the note-book. A regular system of book-keeping is no less important in the obser- 
vatory than in the counting-house. 

8 



58 THE TRANSIT INSTRUMENT. 

67. We thus obtain, in the most perfect way, the clock times 
of meridian passage for any number of bodies whose right as- 
cension is known. And hence the error of the clock being known 
for a number of instants preceding and following any event, the 
error is of course known at the instant of its occurrence. This 
applied with a contrary sign at once transforms the clock time 
at which the phenomenon was observed to true sidereal time ; 
which is usually the end and object of all the uses of a transit 
instrument out of the observatory. 

All this, however, supposes that our instrumental adjustment 
of the transit instrument is perfect, which never is the case. So 
far from it, indeed, that without corrections to be applied for the 
errors of the instrument, the observer with an ordinary tran- 
sit can seldom reckon with certainty on an accuracy greater 
than within 2 s or 3 s , and often much less. And even if no greater 
is desirable, yet some apprehension of the mode of correcting its 
errors is almost indispensable, that he may proceed with that 
knowledge of his means, and that acquaintance with the manage- 
ment of his instrument, so essential to inspire confidence in his 
results. The remainder of the chapter will therefore be devoted 
to a short consideration of the errors of the transit instrument, 
and their corrections. 

68. The errors of the transit instrument and their corrections. 
Let 9 be the latitude of the place. If then 6 represents the 

declination of any body upon the meridian above the pole, nega- 
tive when south, and positive when north, and if when below 
the pole, S equals 180° — declination, it is easily seen that (p—8 
will in all cases represent the zenith distance of the object, con- 
sidered as positive when south, and negative when north of the 
zenith. 

69. Suppose the telescope first to revolve in a plane cutting 
the meridian in the zenith, and passing very slightly to the east 
of it on the south side, and west of it on the north. Such a 
situation may be represented on an artificial globe, by turning 
the pole up to the zenith, so that the equator may coincide with 
the horizon, in which case one of the colures, turned a little from 
the south point towards the east, will represent the plane of our 



THE TRANSIT INSTRUMENT. 59 

transit. Or the same may be done imaginarily with fig. 18, call- 
ing PIP' the meridian. The angle of deviation of the transit 
plane will then be measurable on the southern horizon, which 
call a, or 

Let a = the azimuthal deviation ; + when, measured on the 
southern horizon, it deviates towards the east, and — when to- 
wards the west ; and let it be expressed in time,* and not in 
arc. 

Now the deviation being equal to a at the horizon, will de- 
crease as the two circles converge towards one another, till at 
the zenith it becomes ; or at any known zenith distance, the 
distance between the two circles (Cor. 2, Note, Art. 63.) = a x 
sin zen. dist. —a sin (<p— 5). Now a star south of the zenith in 
coming to the meridian, will cross this small space (Cor. 1, Note, 
Art. 63.) in a time —a sin (p— 6) sec S. If therefore the star ar- 
rives at the transit plane at the time t, it will pass from one circle 
to the other, and come to the meridian at the time 

t-\-a sin (9— 6) sec 8. 
As 9— 6, and of course sin (<p — 6), is negative to the north of the 
zenith, the correction must always become subtractive between 
the zenith and the pole ; and manifestly it ought, for the transit 
plane in that quarter passes over to the west of the meridian. 

70. Suppose again the eastern pivot to be a little lower than 
the western, or which is the same thing, that the telescope re- 
volves in a plane cutting the meridian in the north and south 
points, and turned a little over eastward from the meridian at 
the zenith. This position again may be imitated by bringing the 
poles on a globe or in fig. 18, to coincide with the horizon, and 
turning a colure from the zenith a very little over to the east. 
The error of inclination will then be expressed, of course, by 
the small arc of the prime vertical intercepted, which repre- 
sent by 

b = error of inclination ; + when, measured on the prime ver- 
tical, it inclines towards the east, — when towards the west ; and 
expressed in seconds of time. 

The inclination being = b at the zenith, decreases towards the 

* Any great circle, as the horizon, may be supposed to be divided into hours, mi- 
nutes, and seconds, instead of degrees, minutes, and seconds. 



60 THE TRANSIT INSTRUMENT. 

horizon either way, and the distance between the circles at any 
: point is (Cor. 2, Note, Art. 63.) b x sin altitude = b x cos zen. dist. 
— b cos (9— 8) ; and a star arriving at the transit plane at the 
time t, will cross it, and (allowing for its rate of motion as be- 
fore) arrive on the meridian at the time 

t-\-b cos (9 — 8) sec 8. 
Now cos (9— £) does not change its sign when 9— 8 becomes ne- 
gative at the north of the meridian ; and the reason is easily 
seen, — for the transit plane is in this case wholly to the east of 
the meridian from the north to the south point, or when b is ne- 
gative wholly to the west. 

71. Lastly, the telescope may not move in the meridian, but 
in a small circle parallel to the meridian, and every where a cer- 
tain number of seconds (c) east of it. This will be the case if 
the optical axis, instead of being perpendicular to the horizontal 
axis, is inclined towards the eastern pivot a little. Let then 

c = the error of collimation ; + when eastward, — when west- 
ward of the meridian, and reckoned in seconds of time. 

The distance from this circle to the meridian being constant 
at all points, a star arriving at the transit plane at the time t, will 
pass from one to the other in c sec 8 seconds, and reach the me- 
ridian at the time 

t-\-c sec 8. 
If, during the observations, the transit should be reversed, the 
telescope will point as far to the westward of the meridian as 
before to the eastward, and the star will arrive on the meridian 
c sec 8 seconds earlier than t, that is, at t—c sec <5; c being the 
error in the first position. 

An objection has perhaps occurred to the reader of this kind ; 
that since the stars move in an opposite direction beneath the 
pole, all these three corrections ought to be with changed signs 
in that quarter, as in the reduction to the mean wire, (Art. 65.) 
This inference is a perfectly correct one ; and on looking at our 
formulae, we see that, since 8 is greater than 90° below the pole, 
sec 8 becomes negative in all the foregoing expressions, thereby 
causing the very change in question. 

Now, supposing all these errors to take place at once, since they 
are all small, and therefore in a direction perpendicular to the me- 



THE TRANSIT INSTRUMENT. 61 

ridian, the combined effect of the three at any point is the algebraic 
sum of the separate effects ; consequently, a star, whose recorded 
time of passing the mean wire is t, is on the meridian at the time 

t+a sin (9 — 8) sec 8+b cos (9— 8) sec 8+c sec 8. 
tf <z, b, c, are all + or towards the east, this correction of t is 
entirely additive, as it is plain it should be, since the star moves 
from east to west ; and vice versd,. 

72. Once more, let x = the correction of the clock ; + when 
the clock is slow of sidereal time, — when fast of the same. 

Then t being the clock time of passing the mean of the transit 
wires, t-\-x is the true sidereal time of passing the same, and as 
above . . . t-{-x-\-a sin (9— 8) sec 8 + &c. is the true sidereal time 
of the star's crossing the meridian. But this == its own right 
ascension, which call a. We have therefore 

a == t-\-x-\-a sin (9 — 8) sec 8-\-b cos (cp—8) sec 8-\-c sec S. (1.) 
orx= <x— t— a sin (9 — 8) sec 8—b cos (9 — <5) sec 8—c sec 8. (2.) 

Now, our object being to find the error of the clock at any 
moment, we may do so by this equation, provided we know the 
values of a, b, and c. For all the other quantities are known, a, 
the R.A. of the star ; t, the recorded time of its passage, and 8 
and 9, its known declination and the latitude of the place. 

73. To find a, b, and c. — First, the level furnishes us with a 
direct means of ascertaining the quantity b. The horizontal axis, 
when the inclination is positive, is of course depressed below the 
east point by an angle equal to b ; and the level indicates this de- 
pression, if we know to what arc in space each division of its 
scale corresponds. An accurate method of finding the value of 
a division, if unknown, is illustrated in our example, (Art. 76.) 

Let the axis of the transit be nearly horizontal, and the bubble 

of the level extending on each side of the central zero of the 

scale. In this position, let e designate the reading of the east 

10 — e 
end, and w that of the west end ; then — — represents the num- 

ber of divisions by which the centre of the bubble is west of 
zero.* Let the east and west readings of the bubble in the re- 

* Of course, if — -— is negative, the centre of the bubble must be east of zero. 



62 THE TRANSIT INSTRUMENT. 

versed position be e' and w', then — - — is the distance of the cen- 

tre of the bubble east of zero. A consideration of Art. 36. shows 

that when the axis is horizontal, these two should be equal, 10 div 

W. in the usual position coinciding with 10 div E. in the reversed. 

And if the eastern pivot is depressed by b, expressed in divi- 

e 1 — w/ w — 6 

sions of the scale, — - — must be less than — — by twice that 

[10 ~~ B ) — {g! ~~~ ID i 

amount, or 2b. Therefore 2b = —— ■ -, or more con- 

/it 

(iD-X-y) 1 ) — (e~ l-fij 

veniently b = ^ —~ '. To express b, however, in seconds 

of time, let k be the value of l div of the scale ; then 
t,_7 (w+w') — (e+e ; ) 
4 
If the indications of the level are not constantly the same, or 
very nearly so, the level should be read off in both positions 
several times during observation, and the value of b at those times 
will thence be found with ease by the above formula. 

74. The error of collimation (c) may be found in several ways. 
First, rudely, by making the central wire bisect the meridian 
mark, and after reversion, estimating its displacement in seconds of 
time by a comparison with the known distance between any two 
wires ; one half of this displacement = c, and is positive, if,' in 
an inverting telescope, the wire after reversion appears on the 
left hand of the meridian mark. 

Another and better mode consists in reversing the transit as 

soon as a star has crossed the middle wire, and noting the times 

of its repassing the II and I, or the two which it first crossed in 

the natural position. The time of transit over the mean wire in 

the usual position can then be found from the observed wires I, 

II, and III, by our rule, Art. 65 ; and the time it would have 

crossed the same in the reversed position, from the wires II and 

I again observed. Let these times be t and t' respectively ; then 

both t+c sec S and t'—c sec 5, by Art. 71, express the time of 

the star's arriving on the meridian. They are therefore equal, 

t'—t 

and c= i £ = i (t—t) cos 6. A few repetitions of this pro- 

sec o 



THE TRANSIT INSTRUMENT. 63 

cess in the course of an evening, especially on slow-moving polar 
stars, will give a very correct value of c. 

Frequently, however, a transit instrument is not situated so as 
to command circumpolar stars, and the following method is in 
all cases available, and generally more accurate with ordinary 
instruments. Having observed a number of stars on widely dif- 
ferent points of the meridian, reverse, and observe about as many 
more. Then from each star in the first position we shall have 
an equation like eq. (2,) Art. 72, in which c sec 6 will have the 
sign — before it ; and from each observed after reversion, a 
similar equation, except that c sec <5 will be affected with the op- 
posite sign. The quantity c may then be eliminated, as in our 
example, Art. 76. 

75. The quantity a must be found by a combination of the 
observations themselves, in the form of equation (2,) Art. 72. 

To each star observed, apply eq. (2,) giving to a — t, sin (<p— S) 
sec 5, cos (p— 8) sec <5, sec 8, their respective values. The three 
latter are called the coefficients of a, b, and c. For any given lati- 
tude or place, they may be calculated for every degree, or every 
second degree of declination, both + and — , as far as the range 
of the instrument extends. The calculation of such a table will 
occupy but a small space of time, and saves very much labor in 
reducing observations. Such a table for the latitude of New 
Haven is given in Table V, and any observer may in an hour or 
two fill out a similar one for his own station. By its aid, eq. (2) 
may be formed for each star almost instantly. For example : 

July 17, 1838.— Observed transit of j3 Aquilae . . . 19 h 47 m 19 s . 12. 
By Naut. Aim.— R.A. of (3 Aquilae . . . 19 h 47 m 24 s .22 ; Decl. +6° 0'. 

Therefore a—t = +5M0, and looking in Table V, against Decl. 
+ 6° 0', we have by mere inspection the coefficients of a, b, 
and c, as follows : 

(3 Aquilae . . . a:=+5 s .10 — .58 a— .82 6—1.01 c. 
A similar equation may be as easily formed for each star. That 
x may be the same in all the equations, allow for the known rate 
of the clock in each. Next, the value of b being ascertained by 
reducing the observations on the level, multiply it into its coeffi- 
cient in each equation, and incorporate the same with the cor- 



64 THE TRANSIT INSTRUMENT. 

responding value of a—t. The only unknown quantities are 
now x 9 a, and c. 

If c has been obtained by reversing on several stars, the frac- 
tion of its value may be also annexed to a—t in like manner with 
that of b, and the equations combined so as to obtain x alone. 
But if c likewise is to be obtained from comparison of all the ob- 
servations, we may proceed as follows : 

(1.) Add a number of equations, which contain a large coeffi- 
cient of a, to form one equation, and an equal number of those 
containing small coefficients of a for another ; taking care that 
there shall be the same number of plus as of minus coefficients 
of c in each of the two newly formed equations. Subtracting 
one of these from the other, we eliminate x at once, and, if the 
equations are well selected, obtain a with so large, and c with 
so small a coefficient, that the latter may usually be neglected, or 
allowed for afterwards. Dividing by its coefficient, we obtain 
the value of a. 

(2.) Again, form two other equations, one embracing plus co- 
efficients of c, the other as many with the minus sign, at the same 
time choosing those in which the coefficients of a nearly balance 
one another. By subtracting as before, we have a very large 
coefficient of c, and a very small one of a, and a being already 
known, c becomes so of course. 

The student will remark, that although the unknown quanti- 
ties x, a, c, might be eliminated from any three of the equations, 
yet the results would not perfectly agree with those deduced 
from any other three, on account of errors of observation ; a 
combination of the whole as above recommended is much less 
affected by such errors, and is therefore more accurate. 

Substituting the values of a, b, and c, in the original equations, 
we obtain from each, x, or the correction of the clock at the time 
their respective stars crossed the meridian. If any event, such 
as the occultation of a star, the transit of the moon's limb, an 
eclipse of one of Jupiter's satellites, the firing of a preconcerted 
signal, or the flash of a meteor, has in the mean while taken 
place at a certain time by the clock, a judicious comparison of 
this series of corrections will give a very accurate value of the 
required correction of the clock at the time of this event ; and 
nothing further remains, but to apply it accordingly. 



THE TRANSIT INSTRUMENT. 



65 



76. Example ; an evening's observations, and their reduction. 

We conclude with an example of the observations of a single 
evening, made with an uncompensated clock, and a transit instru- 
ment of but ordinary excellence, which commanded the southern 
meridian up to 70° of altitude. The actual reduction of such a 
series will illustrate and enforce the previous descriptions of pro- 
cesses, and furnish a model for the observer, which he may fol- 
low in whole, or in part, according to the degree of accuracy he 
is desirous of obtaining. 

New Haven, Lat. 41° 18'. Oct. 17, 1839.— The occultation 
of S Capricorni having been observed, — it is required to deduce 
from the transit observations of that evening the true sidereal times 
of immersion and emersion. 

The transit journal furnishes the following observations : 



Objects. 



rj Capricorni. 

s " 

/Pegasi 

"3)'s 1 Limb.. 
6 Capricorni. 
i Aquarii 



£ Pegasi. 



Immersion 



a Piscis Australis. 
a Pegasi 



Emersion 



y Pegasi. 
(iCeti .. 



rj Andromedae. 



17.6 
56.9 
33.9 
50.9 
14.9 



of 

30.9 
V. 



55.3 
44.6 
of 

56.8 
31.8 

31.3 



25.9 
3.0 
43.5 
59.8 
21.7 
57.4 

6 
35.6 
IV. 



0.0 
51.1 

6 

3.4 
39.8 

II. 
40.3 



III. 



| IV. 



h. m. s. 



20 54 34.8 

21 6 9.5 
" 21 53.1 
" 35 8.8 
" 37 28.7 
" 57 3.8 
Capricorni 

22 32 40.8* 
III. 

« << 

" 48 4.5 
" 55 57.3 
Capricorni 
4 9.5 
" 34 47.5 

III. 
" 47 50.2 



43.1 
16.3 

2.8 
17.5 
35.8 

9.9 



II. 

44.6* 

9.0 

3.7 



55.3 
IV. 
59.5 



51.9 
23.1 
12.5 
26.4 
43.1 
16.3 



I. 

49.4 

13.0 

9.9 



3.1 

V. 
9.2 



Reduction. 



h. m. 



20 54 34.66 
" 6 9.76 
" 21 53.16 
" 35 8.68 
" 37 28.84 
" 57 3.71 

22 22 43.80 
" 32 40.98 

LampWest.f 
" 32 39.19 
" 48 4.36 
" 55 57.32 

23 14 42.5 
4 9.50 
" 34 47.50 

Lamp East.f 
" 47 50.10 



At 21h 45 m . 



Cross level E. 
W. 

E. 
W. 

E. 
W. 



Subtract sum of E. 
from sum of W. readings. 



East. 



47.6 
55.5 
48.1 
55.0 
47.0 
54.8 



308.0 



Divide by the number of readings} 12 4* 321 



West. 

61.3 
53.2 
59.7 
53.2 
60.6 
52.1 



340.1 
308.0 



At Oh 10 m . 
Cross level E. 
W. 



48.6 
55.8 


62.2 
53.7 


104.4 


115.9 
104.4 


4 


+ 11.5 



b =+2.88 k 



b=-i- 2.67 k 



* The axis being- reversed during the passage of r, Pegasi, (see Art. 74.) 

t The illuminating lamp in the present case was at the west end of the axis after 

reversion, and serves to distinguish between the two positions. 

X The process indicated in the formula (Art. 73,) is here repeated three times for 

the sake of diminishing small errors, and 12 is therefore the divisor instead of 4. 

9 



66 



THE TRANSIT INSTRUMENT. 



The reductions in the last column are made by the rules in 
Arts. 64 and 65. The apparent right ascensions of the stars 
observed are next to be obtained from the Nautical Almanac, or 
from some catalogue by Problem IV, Art. 81 ; also from the 
same catalogue their declinations to the nearest minute of space, 
allowing for precession. The work will then be as follows : 



Stars. 


a 


t 


a-t 


8 


Equations I. 


?7 Capr. 


20 55 17.68 


54 34.66 


+43.02 


-20° 29' 


a: =+43.02- .94 a- .50 6- 1.07 c 


* " 


21 6 53.40 


6 9.76 


-(-43.64 


-15 


50 


x = +43.64- .87 a- .56 6-1.04 c 

2 


/Peg. 


" 22 42.27 


21 53.16 


+49.11 


+22 


56 


^ = +-49.11— .34 a-1.03 6-1.09 c 

3 


y s 1 L, 


" 25 1.66 


35 8.68 


-10 7.02 


-17 


4 


-10 m 7 8 .02- .89 a- .55 6-1.05 c 


8 Capr. 


" 38 12.39 


37 28.84 


+43.55 


-16 


51 


* = +43.55- .89 a- .55 6- 1.05 c 

4 


t Aq. 


" 57 47.84 


57 3.71 


+44.13 


-]4 


39 


x = +44.13- .86 a- .58 6-1.03 c 

5 


(Peg. 


22 33 29.38 
Axis 


32 40.98 
reversed. 


+48.40 


+10 





x =+48.40- .53 a- .87 6-1.01 c 

6 


5 " 


" 33 29.38 


32 39.19 


+50.19 


+10 





* =+50.19- .53 a- .87 6+1.01 c 

7 


a P. A. 


" 48 48.64 


48 4.36 


+44.28 


-30 


28 


x-= +44.28- 1.10 a- .36 6+1.16 c 

8 


a Peg. 


" 56 48.19 


55 57.32 


+50.87 


+14 


21 


* = +50.87- .47 a- .92 6+1.03 c 

9 


yPeg. 


5 0.94 


4 9.50 


+51.44 


+14 


18 


x =+51.44- .47 a- .92 6+1.03 c 

10 


jSCeti 


" 35 34.43 

Axis 


34 47.50 

returned. 


+46.93 


-18 


52 


x = +46.93- .92 a- .53 6+1.06 c 
ii 


n And. 


" 48 40.47 


47 50.10 


+50.37 


+22 


33 


x = +50.37- .35 a- 1.03 6- 1.08 c 

12 



Art. 75 sufficiently explains the manner of obtaining " Equa- 
tions I." The R.A. of the moon is such as it was when crossing 
the meridian of Greenwich, and therefore no correction can be 
deduced from it. 

The quantity x in each equation now denotes the required 
correction of the clock, at the time of passage of its respec- 
tive star, and may therefore be increasing or decreasing. It 
should be made of a uniform value throughout, by applying 
an approximate correction for the rate. For this purpose we 
have / Pegasi in very nearly the same declination with r\ An- 
dromedae, (both being in the same position of the transit axis ;) 
therefore for 24 h .8-21 h .4 = 3 h .4, the rate = +50 s .37-49 s .ll = 
+P.26, or + S .37 pr. hour. Again, s Capricorni crosses a 
point of the meridian very nearly half-way between <5 Capr. 



THE TRANSIT INSTRUMENT. 



67 



(21 h 
= 



38 m +21 h 58 m 



)-21 h .l,or 



0\7, the rate = 4355 + 441 j -43.64=+0 8 .20, or +0 s .29pr. hour, 

Giving a double weight to the former of these two values, on 
account of its deduction from a longer interval, we have r = 
+0 S .34 pr. hour. 

For the value of b, the observations give us b = +2.67 k, and b = 
+2.88 k, at 2l\8 and h .2 respectively. Being nearly the same, 
we may combine them, giving the first double weight, and b = 
+2.74 k. To determine k, it had been found by a scale and ver- 
nier, that 28 turns of the screw of elevation raised its Y 0.677 
inches ; also 25 turns were equal to 0.605 inches ; .*. 1 turn = 
.02418 by the 1st, and =.02420 by the last measure; therefore 
1 turn =.02419 inches. Distance from Y to Y = 13.57 inches ; 
then 13.57 : rad : : .02419 : arc 6' 7".7 = 367".7 = 24 s .51. And 
a of a turn was found on a mean of many trials to change the 
situation of the bubble, when near its usual limits, by 10.79 divi- 

24 s 51 

sions. Therefore — i— or 3 8 .501 =10.79 div. and l div or k = 
7 

9 .325. Consequently, for our observations, b = + 2.74 k = 
+ 8 .89. 

With the above value of r we will reduce the expressions for x 
to such as they would be at 21\0 ; and also eliminate b from the 
equations. The corrections for r and b are tabulated below, and 
applying them to a— t f we will reduce "Equations I" to "Equa- 
tions II." 



Correction for 




r = +\34 


b = +'.89 




+ .03 


— .45 




r 42 s . 60 


— .03 


— .50 




-43.11 


— .12 


— .92 




- 48 .07 


— .21 


— .49 




- 42 .85 


— .32 


— .52 




- 43 .29 


— .53 


— .77 




-47.10 


— .53 


— .77 




r 48 .89 


— .61 


— .32 




- 43 .35 


— .66 


— .82 




- 49 .39 


—1.04 


— .82 




-49.58 


—1.22 


— .47 




- 45 .24 


—1.29 


— .92 




-48.16 



No. 



Equations II. 



x = + 42-.60 — 
x = + 43 .11 — 
48 .07 — 
+ 42 .85 — 
43 .29 — 
+ 47.10 — 



48 


.89 


— .53 


n — 


43 


.35 


—1.10 


a — 


49 


.39 


— .47 


a - 


49 


.58 


— .47 


a — 


45 


.24 


— .92 


a - 



.94 a— 1.07 c 
.87 a— 1.04 c 
.34 a — 1.09 c 
.89 a — 1.05 c 
.86 a — 1.03 c 
.53 a — 1.01 c 
1.01 c 
1.16 c 
1.03 c 



1.06 c 
f 48 .16 — .35 a — 1.08 c 



By properly combining "Equations II" in the way recom- 
mended in Art. 75, we find the values of a and c as follows : 



68 



THE TRANSIT INSTRUMENT. 



Adding (1.), (2.), (4.), (8.)» (H.)» to form tne nrst equation, 
and (3.), (6.), (9.), (10.), (12.), to form the second, we have, 
5x = +217 s .15— 4.72 a— 0.94 c 
5tf = +242 s .30— 2.16 a— 1.12 c 

Subtracting = - 25M5— 2.56 a+0.18 c 
2.56a = — 25U5 + .18 c 
a = — 9 S .82 + .07 c 
Again, 
Adding (7.), (8.), (9.), (10.), (11.), 

5 x = +236 9 .45— 3.49 a + 5.29 c 
Adding (2.), (4.), (5.), (6.), (12.), 

5a? = +224 s .51-3.50a— 5.21c 



= + 1P.94+ .01 a+ 10.50 c 
10.50 c= -1P.94— .01a = -ll s .94+0U0 = — 11.84 
c = — 1M3 
Correcting the value of a as above by this of c, 

a = -9 S .82— S .08 = -9 S .90. 
Finally, we substitute the values of a, b, and c, in " Equations 
I," and thus obtain " Equations III," or the corrections of the 
clock. 



Equations III. 


X 


x = -\- 43.02 -J- 9.31 — .45 -j- 1.21 = 


-f-53 s .09 


x = -f- 43.64 -J- 8.61 — .50 + 1.18 = 


+ 52 .93 


x = -f- 49.11 -+• 3.57 — .92 + 1.23 = 


+ 52 .79 


x = + 43.55 + 8.81 — .49 + 119 == 


+ 53 .06 


x = -f- 44.13 + 8.51 — .52 + 1.16 = 


+ 53 .28 


x = + 48.40 -f 5.25 — .77 -f- 1.14 = 


+ 54 .02 


x — + 50.19 + 5.25 — .77 — 1 .14 = 


+ 53 .53 


x = + 44.28 -f- 10.89 — .32 — 1.31 = 


+ 53 .54 


x = + 50.87 + 4.65 — .82 — 1.16 = 


+ 53 .54 


x — -\- 51.44 + 4.65 — .82 — 1.16 = 

10 


+ 54 .11 


a? = *-t- 46.93 + 9.11 — .47 — 1.20 = 


-j- 54 .37 


x = + 50.37 -f- 3.46 — .92 -j- 1.22 = 

12 


+ 54 .13 



"We see that x, the correction of the clock, is, allowing for the 
errors of observation, nearly stationary at first, but afterwards 



x = + 52 8 .97 

™ .... +53.47 

.... +54.20 



+ .50 

+ .73 



PROBLEMS. 69 

gradually increasing. The result on the first three wires of £ 

Pegasi, not well agreeing with the rest of the series, should be 

cast out ; then taking the mean of the first four, and of their 

corresponding times, and the mean of the next four, as also of 

the last three, we have 

at 21 h 15 m 

22' 1 33 m 

h 29 m 

Hence for any given time t' between 21 h 15 m and 22 h 33 m 

s *so 

x = +52.97+ (t'-21 h 15 m ) ~jp 

and between 22 h 33 m and h 29 m 

O 8 7*3 

x = +53.47+ (t'-22 h 33 m ) — — . 
v J 116 

s 50 
Then at 22 h 23 ra x = +52.97+68 ~~ = +53 8 .41 

s 7S 

at 23 h 15 m x = +53.47+42 ~— = +53 8 .73 

116 

Im. of 8 Capricorni 22 h 22 m 43 s .80+53. 8 41 = 23 h 23 ra 37 8 .21, 

Em. " 23 h 14™ 42 8 .5 + 53 8 .73=23 h 15 m 36 8 .2, 

true sidereal time at New Haven. 



CHAPTER V. 

OF PROBLEMS IN PRACTICAL ASTRONOMY. 

77. In the calculations connected with practical astronomy, 
there are a number of operations so frequently required, that 
familiarity with at least a few is necessary to render farther pro- 
gress easy or certain. The following are some of the most use- 
ful and universal in their application. 

I. On the use of Signs connected with the Logarithms of Numbers 
and Circular Functions. 

78. The sign + or — prefixed to a logarithm indicates the 
state of the natural number to which the logarithm belongs^ and 



70 PROBLEMS. 

not of the logarithm itself. Thus, in the operations of multipli- 
cation and division, the signs belonging to quantities are pre- 
fixed to their respective logarithms, and being disregarded in 
the addition or subtraction of the latter, are only of use to 
determine what sign shall be prefixed to the final product or 
quotient. 

The sine, cosine, tangent, &c. of an arc, it will be recollected, 
have not the same signs in all the quadrants. The sine, for in- 
stance, is positive in the 1st and 2nd quadrants, but in the 3rd 
and 4th, being measured in an opposite direction, it is considered 
negative. For convenience of reference, Table I. contains a 
schedule of the changes of the different circular functions through 
the four quadrants, with their opposite signs in each. In com- 
mon operations of trigonometry and astronomy, the change of 
signs caused by difference of quadrant may frequently be neg- 
lected ; it is, however, the best, because the safest and most uni- 
form method, to retain the distinction in all cases. 
Ex. What is the value of 

(— 627".l)x sin. (-28° lOQx cos (-28° 10') 

cos 157° tan 98° sin 30° 

Remark. — As the sine of +28° 10' is the same as that of 360° 

+28° 10', so the sine of -28° 10' is the same as of 360° -28° 10'. 

The sine and cosine of —28° 10', are therefore those of the 4th 

quadrant. 

Z. (-627".l) -2.7973 

I sin (—28° 10') -9.6740 

/.cos (-28° 10') +9.9453 

Z. cos 157° (a. c.) -0.0360 

Z. tan 98° (a. c.) -9.1478 

Z. sin 30° (a. c.) +0.3010 

+ 1.9014 + 79".7 

II. On the conversion of Solar into Sidereal Time, and the con- 
trary. 

79. The sidereal day is shorter than the solar, and renders 
necessary for the comparison of portions of either with the 
other, a " Table of Equivalents." Such are afforded in Tables II 
and III. 

The sidereal time corresponding to mean noon at Greenwich is 



PROBLEMS. 71 

given for each day in the Nautical and American Almanacs, and 
usually in all ephemerides of any importance. Also in the Naut. 
Aim. is given the " Mean Time of Transit of the first point of 
Aries," or the mean time corresponding to sidereal noon. The 
sidereal time therefore corresponding to any hour, minute and 
second of Gr. mean time, is found by adding to the sidereal time 
of the preceding Gr. mean noon the sidereal equivalent of the 
mean time since then elapsed ; and vice versa. The reason of 
the process will perhaps be best understood by expressing the 
agreement between mean solar and sidereal time in the form of 
an equation. 

Ex. 1. To convert 13 h 41 m 17 8 .04 Gr. mean time, July 1, 1840, 
into Gr. sid. time. 

Gr. mean time. Gr. sid. time. 

July 1, 1840. h m S .00 = 6 h 38 m 24 s .89 ... p. II. N. A. 
13 h =13 2 8.13 ) 

41 m =416 .74 > Table II. 

17 3 .04 = 17.09 \ 



13 h 41 m 17 s .04 = 20 h 21 m 56 s .85 
Ex. 2. To convert 3 h 10 m 19 3 .76 sid. time into mean solar time, 
July 1, 1840, for the meridian of New Haven. 

N. H. sid. time. N. H. mean time. 

17 h 18 m 44 s .48 I Gr.m.t. ofsid.noon. 
- 47 .80 \ p. XXII. N. A. 



Transit of 1st 
point of Aries, 



J h Q m s .00=17 17 56.68 N. H. " 

3 =2 59 30.51 ) 

10=9 58.36 V Table III. 

19.76= 19.71 ) 



3 10 19.76=20 27 45.26 
Remark. — At any other meridian than Greenwich, — for in- 
stance, at New Haven, since we have no equation between mean 
and sidereal time at New Haven mean or sidereal noon, we con- 
vert New Haven into Gr. time by adding the difference of lon- 
gitude, then convert Gr. mean into Gr. sid. time, or Gr. sid. into 
Gr. mean time, and again subtract the longitude to reconvert to 
New Haven time. These operations may be materially shortened, 
by adding 47 3 .93, or " the acceleration of sidereal time on solar" 
during 4 h 51 m 46 9 (Table II.), to the Gr. sid. time of Mean Noon, 
to find the N. H. sid. time of Mean Noon ; and by subtracting 



72 PROBLEMS. 

47 8 .80, or " the retardation of solar time on sidereal" during 4 h 
51 m 46 s (Table III.), from " Mean Time of Transit of First 
Point of Aries," as given for Greenwich, to find the same for 
New Haven. 

Some ephemerides give only " Sid. Time at Mean Noon ;" in 
this case, Ex. 2 may be performed by reversing the operation in 
Ex. 1. For instance, to find the mean time answering to 20 h 
21 m 56 s .85 sid. time, take out from Table III. the equivalent of 
20 h 21 m 56 3 .85-6 h 38™ 24 s .89 in mean time. 

It should be always remembered, that all the quantities in the 
Nautical Almanac are in Greenwich time. The local time of any 
other place must therefore be invariably converted into the cor- 
responding Greenwich time, before any reference can be made 
to this ephemeris.* 

III. Interpolation by Differences. 

80. It is frequently necessary in Practical Astronomy, from 
the values of any variable quantity at certain equal definite in- 
tervals, to determine its amount at any intermediate time. The 
most irregularly varying quantity, if of gradual change, and 
subject to the control of any, even an unknown, law of progress, 
may be concluded from a regular series of values, for any other 
point amidst the series by the method of differences. (See Day's 
Algebra, p. 273.) 

If, for example, we have the moon's right ascension at mean 
noon for a number of days in succession, subtract the R.A. at 
each noon from that at the following noon, paying attention to 
the signs ; and of these first differences subtract each from the 
next below it for a second order of differences, — and so on. 
Call the 1st R.A. in the series «, the 1st of the first differences d', 
the 1st of the second differences d", &c, then for any time t in 
the series, we have (Day's Algebra, Art. 493 .f. — substituting 
t+l for n,) 

A = a+d't+d"t ~ +d'"t ti. ^?+ &c. 

which is the general formula for interpolation, where a repre- 

* The " Explanation of the Articles" in each Naut. Aim., contains in its first 
six pages an exposition of this subject, to which the student may refer with great 
advantage. 



PROBLEMS. 



73 



sents the first of a series of values, d', d", d'", &c, the first of 
each successive order of differences, and A the value for the time 
t elapsed since the time of the first value, and reckoned in parts 
of the equal intervening intervals. 

Ex. 1. Given the sun's declination as follows : 

d' 



1838. 



Sept. 17, 
18, 
19, 
20, 
21, 



-1396".l 
-1398 .7 
-1400 .9 
-1402 .6 



d" 
-2".6 
-2 .2 
-1 .7 



d"' 
+0".4 
+0 .5 



d"" 
+0".l 



At Mean Noon. 

+2° 20' 14".3* 

1 56 58 .2 

1 33 39 .5 

1 10 18 .6 

+0 46 56 .0 

Required the same at Sept. 18 d 9 h 30 m . 

Here, after taking the differences as above, a = +2° 20' 14".3 ; 
- 1396".l ; d"= -2".6 ; d'"= +0".4, &c, and t = l d 9 h 30 m 
1.4 nearly. 

1396".l x lif - 2".6 x 1.4 x ^ + 



119 



A = +2° 20' 14".3 
0.6 



&c. = +1° 47' 44".9 



0.4 

0".4xl.4x — x 

When the series is so rapidly convergent, as to render calcu- 
lation beyond the second differences unnecessary, the interpola- 
tion may be performed by means of Table IV., which contains 
the coefficients of d" for every hundredth part of the unit of 
time elapsing between the successive values of the quantity re- 
quired. The following rule will in this case be applicable : 

" Take from the ephemeris the two values preceding, and the 
two following the required time. Subtract the first in order of 
time from the second, the second from the third, &c, for the 
first rank of differences. Subtract each of these from the fol- 
lowing, for the second rank of differences, always paying atten- 
tion to the algebraic signs of the quantities. Call the 2nd of 
the four values, a ; the 2nd of the three first differences, d', and the 
mean of the two second differences, d". Reduce the time elaps- 
ed since the date of the second value, to the decimal fraction of 
the unit of time chosen, calling the same t. 

" Enter Table IV with this decimal at the side, and d" at the 
top, in the column headed ' Seconds of Second Differences,' and 
take out the corresponding correction, which must have its sign 
always contrary to that of d". Or if d" be greater than 100, 

* North declinations are positive ; south declinations negative. 
10 



74 



PROBLEMS. 



take out its natural or logarithmic coefficient from the proper 
columns, and multiply by d". Add the correction so found to 
a+td' ; the sum will be the value required." 

It is not unfrequently the case, that we have but a limited num- 
ber of values of the required quantity, and wish to interpolate 
for a time not included between them. In such a case we may 
extend the series artificially by addition of differences, until it 
will embrace the time for which we intend to calculate. 

Ex 2. In the calculation of the eclipse, Art. 103, we have only 
the value of p— u included between the lines as below, for 8 h 30 m , 
9 h 30 m , 10 h 30 m ; required the values of the same for 8 h m and 
10 h 45 m . 



7 h 30 r 

8 h 30 r 

9 h 30 r 

10 h 30 n 



p—u 
— 2206 ,/ .3 
-1220 .7 
- 79 .0 

+ 1218 .8 



Here a= — 2206".3 ; d'=+985".6 
a+d't = a+~ = -lllS ,/ .5 



+ 156.1 
+ 156.1 
+ 156.] 
+ 156.1 
+ 156.1 
d"=+156".l ; £=.50 or 30 m * 



+ 985.6 
+ 1141.7 
+1297.8 
+ 1453.9 



Corr. for +100" 
+ 50 
+ 6 



12'.5 
6 .3 

.7 



Table IV. ; Arg. at side. 
= 30 m 



— 1733 .0 
Again, a = +1218.8; d' = +1453".9; d"=+156".l; t = 15 m * 

a+d't = +1582 // .3 
Corr. for +100" — 9.4 ) 

50" - 4.7 £ Arg. 15 m . Table IV. 

6" — .6 ) 



+ 1567.6 
A very useful application of the formula for interpolation is, — 
to find from a series of values of any quantity, the time when 
that quantity arrives at a maximum or minimum, as well as its 
amount at that time. By differentiation of the original formula, 
we have, for the moment that A is at a maximum, this equation : 



d'+d" 



2t-l 



+ &c. =0 



* Where the unit is an hour, the argument at the side may be found expressed in 
minutes. 



PROBLEMS. 75 

d"-2d' 
whence *==_— 

and with t thus found, we may easily obtain the greatest or least 
value, as for instance, of the sun's declination at the solstices, or 
the shortest distance between the centres of any two celestial 
bodies in case of an eclipse or near approach. 

IV. Corrections of the places of Fixed Stars. 

81. All well arranged catalogues of the fixed stars contain 
their mean right ascensions and declinations at a given epoch. 
In instrumental observation, however, we see them not in these 
mean places, but altered by the amount of their precession since 
the given epoch, and affected by nutation and aberration. The 
mean places must therefore be reduced to the apparent, before 
they can be compared with observations. 

The algebraic expressions for these corrections have been so 
subdivided by a distinguished German astronomer,* that all the 
quantities relating to the places of the stars are expressed by four 
factors, a, b, c, d, accented and unaccented ; and all those de- 
pendent on the time, by four others, A, B, C, D, such that the 
whole correction for the place of the star is as follows : 
Aa = «A+fiB+cC+dD. 
A^a'A+ft'B+c'C+d'D. 
where A a and A 5 are the required corrections in R.A. and 
Decl. respectively. 

The arrangement possesses this peculiar advantage, — that the 
factors, expressed by the small letters, being dependent on the 
places of the stars only, and therefore constant for a long period 
of years, may be calculated and inserted with the stars in a stan- 
dard catalogue, while A, B, C, and D, which are the same for all 
the stars, but vary with the time, can be appended to the Nau- 
tical Almanac and similar series of tables, which have the element 
of time instead of space for their basis. The later catalogues, 
such as Schumacher's Catalogue of 500 stars in Pearson's Astron- 
omy, — the Astron. Soc. Catal. of 2900 by Baily, and others, fur- 
nish the logarithmic constants, a, b, c, d, and a', b', c', d', for each 
star ; and the " XXII. p. of the month, Naut. Aim.," contains the 
logarithms of A, B, C, D, for each day. The mean R. A. and 

* Prof. Bessel. 



76 



PROBLEMS. 



Decl. for the epoch of the catalogue is reduced to that for the 
current year, by adding as many times the annual precession in 
R.A. and Decl., annexed to each star in the catalogue, as the 
number of even years elapsed since the given epoch. 

Ex. 1. What is the apparent right ascension of y\ Andromedce 
on the 11th Oct. 1839? 

In the General Catalogue of the Royal Astron. Society, (Mem. 
Ast. Soc., vol. II.) the 96th star is v\ Andromedse, — p. lx, — and 
we there find its " mean R.A. for Jan. 1, 1830," its " annual pre- 
cession," and the " logarithms of a, b, c, and d" as below : 

No. 96. — 38 7} Andromedce. 

LOGARITHMS OF 



a 
+ 8.8486 
+ 1.2322 



0.0808 
+ P.204 



b 

+ 8.1774 
+ 0.9171 



9.0945 
+U24 



+ 0.5029 

+ 9.9389 



d 
+ 8.4315 
-0.9621 



0.4418 

+2 S .766 



Ast. Soc. Cat. 
N. A. p. XXII. 
9.3936 
s .248 Nat. numbers. 



R.A. Jan. 1, 1830, h 48 m 7 S .98 
+28 .647 
+ 1 .204 
+ .124 
+ 2.766 
- .248 



Ann. Prec. 

+3M83 
9 


+28 .647 



h 48 m 40 8 .47 . . App. R.A. of v\ Andromedse. 

The apparent declination may be calculated in the same way 
by employing the factors a\ b', c' , d'. 

" The Apparent Places" of a select hundred fixed stars calcu- 
lated for every tenth day of the year, are contained in the latter 
part of the Naut. Aim. If, of the objects obser pd by a transit 
or other instrument, any should be in this list, or among the four 
" Moon Culminating Stars" to be observed with the moon on 
that evening, — its apparent right ascension may be taken out with- 
out farther trouble ; all others must be computed from the cata- 
logues. 



ECLIPSES OP THE MOON. 77 



CHAPTER VI. 



OF ECLIPSES OF THE MOON. 



82. In calculating an eclipse of the moon, let us conceive our- 
selves encircled by a sphere, concentric with the earth, having 
its radius equal to the moon's distance, and of course, passing 
through the centre of the moon. The minute portion of this 
sphere occupied by the sections of the moon and the earth's shadow 
during a lunar eclipse, may without material error be regarded 
as a plane, and is called the plane of projection ; and the great 
circles in which the centres of these sections move, will be straight 
lines in this plane. If the student has reviewed the general 
method of calculating a lunar eclipse, for which we would refer 
to Arts. 245-260 of Olmsted's Astronomy, he will recollect, that 
the cross sections of the earth's dark shadow and of the penum- 
bra, are two concentric small circles of the sphere, whose semi- 
diameters are respectively P-K— D and P-K+D, where D is 
the sun's semidiameter, P his horizontal parallax, and * that of 
the moon. 

83. Methods of finding the places of the sun and moon in lunar 
and solar eclipses. 

The moon's mean place is the place she would occupy, if she 
revolved around the earth in a circle at her mean distance, and 
undisturbed by the attraction of the sun and planets. The true 
place of the moon differs from the mean place on account of the 
inequalities and perturbations of her motion. All the forces 
which tend to urge the moon from her mean orbit in various 
directions, may be resolved into such as shift her place in the di- 
rection of some great circle, and in a direction perpendicular to 
it. The effects of these forces in changing the moon's mean 
place, are expressed in equations, determined by analysis, and 
containing as factors the mean place of the moon, the place of 
her node, and the like. The numerical results of these equa- 
tions are called the corrections of the moon's mean place, and 
may be computed directly for any particular case, or they may 



78 ECLIPSES OF THE MOON. 

be calculated for every degree of the great circle, and regis- 
tered in separate tables. These Tables being prepared, all that 
is necessary is to take from them the several corrections requir- 
ed, and measure them off backwards and forwards according to 
their signs on the great circle employed, to arrive from the mean 
to the true place on that circle. The corrections for the moon's 
place on a perpendicular circle may be applied in the same way. 
If this calculation be made for regular intervals of time, such 
as mean noon of every day, we shall have a new series of Tables, 
depending on the former, as the former do upon the original ana- 
lytical equations. There will be a distinctive difference between 
them in this respect : — that since the former are computed for 
every degree in a great circle, their element is space, and they 
are complete as soon as the round of the circle is made ; the lat- 
ter, on the other hand, having time for their basis, are in their 
nature continuous, having no definite end. The first class of 
tables are the common Lunar Tables.* The latter are those 

* Many of the older works treating of eclipses, and some of the more recent, in- 
clude an abridged series of Lunar Tables, which have been omitted in the present 
treatise. The pages of the Nautical Almanac have been preferred as Tables of re- 
ference, and for the following reasons : The process of computing the moon's true 
place from the Lunar Tables is not necessarily connected with the calculation of an 
eclipse, any more than is the process of computing the Lunar Tables from the origi- 
nal equations. It is moreover of little profit to the student, since after a knowledge 
of the method of procedure as described above, he has only to enter a great number 
of tables in succession, and take from them their respective additive or subtractive 
corrections, which yet are so numerous, as to render the operation a very long and 
tedious one. This labor, before the wide diffusion of the Nautical Almanac, was in- 
deed indispensable. But since that work is every where and easily obtainable, as- 
tronomers no longer regard this prefatory calculation as essentially connected with 
the work of an eclipse ; and in the later European works on Practical Astronomy, 
instead of giving rules for obtaining the corrections from the " Lunar Tables" for the 
true place, they give rules for interpolating the true place for any given instant of time 
from the pages of the Nautical Almanac. 

It is scarcely necessary to remark, that although the calculation by latitudes and 
longitudes had some advantages while the Lunar Tables were employed, the use of 
the Nautical Almanac renders that by right ascensions and declinations preferable. 
For in that work the moon's longitude and latitude are given only for every noon 
and midnight ; while her right ascension and declination are given for every hour 
of the year, and we thus dispense with interpolation by second and third differences. 
Moreover, although in a lunar eclipse the same formulae are equally applicable, whe- 
ther the equator or the ecliptic be the circle of reference, — yet in a solar eclipse, the 
fixity of the pole of the equator, as regards the zenith, whence parallaxes are reck- 
oned, renders the calculation by R.A. and Decl. more simple and more expeditious. 



ECLIPSES OF THE MOON. 79 

comprised in the Nautical Almanac,* and upon these we shall 
depend for our data in the subsequent calculations. 

The sun's place, the moon's semidiameter and horizontal paral- 
lax, and other elements necessary to eclipses and astronomical 
computations in general, are also more easily, directly, and accu- 
rately obtained by proportion or interpolation between the quan- 
tities in the Naut. Aim., than from the ordinary tables. 

84. Demonstration of formula. 

Let A and A be the true R.A. and Decl. of a point in the 
sphere diametrically opposite to the sun's place, that is, of the 
earth's shadow, and let a and 5 represent the same quantities for 
the moon ; aslo let T be the even hour nearest the middle of the 
eclipse. The quantities A and z/ are the sun's R.A. and Decl., 
(pp. I— II, N. A.,) the first increased by 12 h , the latter with its 
sign changed ; and a and 5 are given for every hour of mean 
time, (pp. V-XII. N. A.) T is readily found for any eclipse by 
comparing the columns of the sun's and moon's R.A. ; a simple 
inspection will show to the nearest hour of mean time when these 
are 12 h distant from each other. Then take out the quantities 
A, J, a, and <5, for the times T— 1'\ T, T+l h ; the two first by 
interpolation, the two last directly. 

In fig. 20, let ACD be the section of the earth's shadow, ASD 
the meridian passing through its centre, and CSF a great circle 
perpendicular to it. Let EMB be the relative orbit of the moon, 
the earth's shadow being supposed stationary ; then a— A, and 
6 — -J, will represent the relative differences of R.A. and Decl. 
of the moon from the centre of the shadow. The section of 
the moon and shadow being regarded as a plane, in which the 

* The Nautical Almanac is a work, which in the present state of astronomy, no 
observer can easily dispense with, even for the simplest uses of the transit instrument 
and sextant. For the purposes of a class in college, who wish to calculate an eclipse 
of the moon or sun in advance, a single copy in the library of the institution will 
suffice; and the few data that are required may be transcribed in a few minutes. 
The work may generally be obtained for at least three years in advance in our large 
cities. 

For most of the purposes indicated in the present treatise, and for all connected 
with eclipses, " Blunt's edition of the Naut. Aim." will suffice. Either this Abridg- 
ment (at $1J)) or the larger London edition (at $2£) can always be obtained in ad- 
vance by application to the agents, E. & G. W. Blunt, New York. 



80 ECLIPSES OF THE MOON. 

perpendiculars AD and CF are two axes of reference, — we must 
now ascertain the moon's perpendicular distance in seconds 
of space from AD and CF at the three times T— l h , T, and 
T+l h . Now, (Cor. 2, Note, Art. 63,) the distance of the moon's 
centre from the meridian AD = (a— A) cos 6. Its distance from 
CF is very nearly equal to 5— J, and would be quite so, but that 
the great circle CF does not coincide with the small circle, paral- 
lel to the equator at that point, for any considerable distance, 
but departs from it by a minute quantity increasing as the square 

of (a— A.) This small difference = — - cos 8 sin A (a— A) 2 * 

where r = the radius of a circle, expressed in seconds of the 
circumference = 206265". 

Let then p = (a— J) cos 6. 

q = § — j -j- __ cos 8 sin d (a— A) 2 
2r v ' 

These are the co-ordinates of the moon's centre with reference 

to CF and AD, and may be measured on those lines. 

85. Take out by simple proportion from the Naut. Aim. the 
sun's semidiameter D and horizontal parallax P, and also those 
of the moon d, if, (pp. II— III. N. A.) Since the earth's shadow 
is nearly elliptical, being the shadow of an ellipsoid, <k (which is 
in different latitudes proportioned to the earth's radius), should be 
reduced from the equatorial value to its value in lat. 45°, by apply- 
ing the reduction contained in Table VI. For the adoption of 
the mean radius of the earth's shadow, will render the calcula- 
tion somewhat more accurate than if its radius, where widest 
across, had been assumed. 

The semidiameters of the dark shadow and of the penumbra 
must be increased on account of the earth's atmosphere. The 
sun's light is partly absorbed, and partly deflected by this medi- 
um, and the effect is much the same as if the earth's radius was 
made a little greater. Astronomers have therefore agreed to in- 
crease the semidiameters of the shadow and of the penumbra 
by eV- Let us call these increased semidiameters s and s' ; 
then 

* For the manner of obtaining this correction, see the chapter on " Occupations 
and Eclipses of the Sun," Art. 99. 



ECLIPSES OF THE MOOHT. 81 

S '=U(P+*-D)+2D. 

Now, when the moon just touches the dark shadow externally, 
as also when she is just quitting it, her distances, BS, ES, from 
the centre of the shadow == s+d ; and at the beginning or end 
of total eclipse, her distance B'S or E'S — s — d. The lines p 
and q, or the co-ordinate distances of the moon from AD and 
CF, evidently form at these times with s dzd a right angled tri- 
angle, of which s± d is the hypothenuse ; and hence at these 
times we have the equation 

(p 2 +q 2 ) 1 2 = sdr.d. 
where s' may be put for s, when the contacts are those of the 
penumbra. 

Now from the series of values which we have of p and q cor- 
responding to T — l h , T, and T+l h , we might find, by a method 
of " trial and error," at what times the first member of the above 
equation should be equal to the last, and these times are, of course, 
those of beginning and end of partial and total eclipse. But in 
the present case, a more direct method presents itself. 

86. If XIII, XIV, XV, in the figure represent the position 
of the moon at the times T— l h , T, T+l h , — then XIII G repre- 
sents the change of p, and G XV that of q, in two hours ; and 
XIII G : G XV : : rad : tan G XIII XV ; which angle, being 
the inclination of the relative orbit BME to the equator, or to 
its parallel CSF, call ». Let t be the time when p becomes no- 
thing, found by simple proportion, and q' be the value of q at 
that time = SN. Draw RSU at right angles to BE, cutting it in 
M ; then since MSN = », MS = SN cos i, and NM = SN sin i. 
Farther, XIII G x sec » = XIII XV, or the motion of the moon 
in two hours, and dividing this by 2, we have h, the horary mo- 

NM 
tion of the moon on its relative orbit BME. Then 60 —j- = 

h 

NM 
the time in minutes of passing from N to M, and Z±60 — j- =t', 

the time of central eclipse, and MS, the shortest distance, is already 
known. In the right angled triangle BMS, we have BS and MS 

11 



ECLIPSES OF THE MOON. 



known, to find BM = v/BS 2 -MS 2 = n/(BS+MS) (BS-MS). 
Then 60 —j- = time of passing from B to M or from M to E, 

and t'=p60 —5— — the times of beginning and end of partial 

eclipse. So also for the beginning and end of total eclipse, and 
for the first and last contacts with the penumbra, we may em- 
ploy s—d and s'+d instead of s+d = BS. 

With this general illustration, the following more distinct for- 
mulae will be easily understood. 

Let i — the inclination of the relative orbit to the circle paral- 
lel to the equator, being positive when 6 is increas- 
ing. 
Ap = h the change of p from T-l h to T+l h . 
A q = " " q " " 

q' = the value of q when p becomes nothing. 
t = the time when p becomes nothing. 
t' = " of central eclipse. 

m = MS, the shortest distance. 
h == moon's horary motion. 

Then (1.) — - = tan i ; » to be taken between 0° and 90°, and + 

or — according to the sign of its tangent. 

(2.) A p sec i = h. 

(3.) dzq' cos i = m ; m to be always positive. 

o 1 sin t 
(4.) t——-r — —t f ; where the time is to be expressed in 

decimal parts of an hour. 
Then the times required are expressed by 
f _^_ s/{sdczd+m) (sdzd-m) ^ 

In the last expression, of the double sign between t' and the 
fraction, — is required in the first contacts, and + in the last. 
Of the double signs under the radical, + belongs to external, and 
— to internal contacts, and if the contacts with the penumbra 
are calculated, s 1 must be substituted for s. 



87. Example. Eclipse of the moon in Feb. 1841. 

For the application of these formulas, let us take as an exam- 



ECLIPSES OF THE MOON. 



83 



pie the eclipse of Feb. 5, 1841, and proceed to calculate the times 
of its different phases for New Haven. And since the absolute 
times of the contacts are the same the earth over, we will first 
calculate them in Greenwich mean time, and reduce to New Ha- 
ven mean time afterwards by subtracting the difference of lon- 
gitude. 

By comparing the columns of R.A. of the sun and moon, we 
see that on Feb. 5, the moon's R.A. at 13 h , 14 h , 15 h is roughly 9 h 
16 m , 9 h 18 m , and 9 h 21 m respectively; while the sun's R.A. on 
the noons of the 5th and 6th is 21 h 16 m and 21 h 20 m respectively. 
Changing the 21 h to 9 h , — since 14 h of time is but little more than 
half-way from noon to noon, the opposition must take place near- 
est Feb. 5 d 14 h , which therefore = T. 

Take from p. II the sun's R.A. for Feb. 5 d — 13 h , 14 h and 15 h 
of mean time* by interpolation, and add to each 12 h ; and from 
pp. V-XII that of the moon for the same times, directly. In the 
same way take out their declinations, changing the sign of that of 
the sun. Combine these quantities so as to obtain the co-ordinates 

(a— A) cos <5 and 6—4+ — cos 6 sin A (a— A) 2 . The latter 

term, being of very small amount, may be roughly computed 
with logarithms extending to only two or three places. The 

logarithm of — is constant, being in all cases 4-4.385. 



2r 



Date. 
Feb. 5d 13^ 
14 
15 



I. a- A 
I. cos 8 
I. 15" (=!•) 



A. 


a 


a-A 1 


9 h 18™23 s .4 
" " 33.4 
" " 43.4 


9 h 16 m 4 s . 3 
" 18 27.1 
" 20 49.4 


-139 s .l| 
- 6.3 
+126.0 



+15° 42' 43" 
+ " 4157 
+ " 41 11 



+16° 2' 23" 
+15 47 55 
+15 33 22 



8-A 
+19' 40" =+1180" 
+ 5 58 = + 358 
- 7 49 = - 469 



l.p 
P 



13'' 
-2.1433 

+ 9.9828 
+ 1.1761 



3.3022 
2005" 



l 4 h 

- 0.7993 
+ 9.9833 
+ 1.1761 



1.9587 



15 h 

+ 2.1004 
+ 9.9838 
+ 1.1761 



+ 3.2603 



91". + 1821' 



2r 





13h 


14ft 


15b 


2 I (a-A) 

I. cos 8 
2 I 15" 


+• 4.29 
+ 9.98 
+ 2.35 


+ 1.60 
+ 9.98 
+ 2.35 


+ 4.20 
+ 9.98 
+ 2.35 


l» 


+ 4.38 


+ 4.38 


+ 4.38 


I. sin A 


+ 9.43 


+ 9.43 


+ 9.43 




+ 0.43 


+ 7.74 


+ 0.34 


A (a-A)* 


+ 3" 


+ 0" 


+ 2" 


8-A 


+1180" 


+358" 


-469" 


2 


+1183" 


+358" 


-467" 



* If the American edition of the N. A. be used, which contains at present the 
sun's R.A., only for apparent noon, the interpolation must be performed for the ap- 
parent times corresponding to 13 h , 14 h , and 15 h of mean time, for which is given the 
equation of time on the same page. 



84 



ECLIPSES OF THE MOON. 



Then at 15*1+1821" 
14 - 91" 



14h 



1912" : lh : : -91" : 0>>.048 = 2 m 53i 



14 h 2* 53* = t 



Again at 14h 1+358" 
15 -467" 



1^ : -825" : : 0^.048 



+358" 
- 40" 

+318" = q>. 



15 h l+1821" 
13 1 - 2005" 


2) +3826" 


&p = + 1913" 


9 
15h|_ 467'/ 

13 1+1183" 


2) — 165G" 



I. A q - 2.9165 
I. A p + 3.2817 

I. tan t - 9.6348 

l = -23°20' 

I. sec i +10.0371* 
I. cos i + 9.9629* 
I. sin i - 9.5978* 



- - - + 3.2817 

Z. sec i +10.0371 

I. h + 3.3188 

h = 2084" 



Z. (±5') ±2.5024 

I. cos t +9.9629 

l m 2.4653 

m = 292" 



A 5 = — 825" 

- - - +2.5024 

7. sin t -9.5978 

I h (a. c.) +6.6812 



14*» 2 m 53* + 3 1 " 36 s = 14 h 6« 29 s 



5' sin t 

- t'. 



8.7814 
Oh.060 



Now, by proportion between the quantities in the N. A., p. II- 
III, we have for Feb. 5 d 14 h , 



Moon's eq. nor. par. = 60' 35" - - 60' 35" 

Sun's " " — 9" 6" - - Reduction to 45° Lat. Tab. VI. 


Moon's semidiam. = 16' 30" tt = 60' 29" 
Sun's " = 16' 14" P = 9" 


s = 45' 8" 
2D = 32' 28" 


60' 38" 
D = 16' 14" 


*' = s+2D = 77' 36" 
d = 16' 30" 


P+r - D = 44' 24" 
i (P-Hr-D) = 44" 


s'+d = 94' 6" 

s = 45' 8" 
d = 16' 30" 


*=|j(P+,r-D):=45' 8" 


s-\-d = 61' 3S" 
s -d = 28' 38" 






For partial eclipse. For total eclipse. 
1 3406" - - +3.5322 I. 1426" H3.1541 


m = 4' 52" 


I. 3990" - - +3.6010 I. 2010" - - - 

2)+7.1332 2)- 

+3.5666 
I. h +3.3188 - 


f3.3032 


s-\-d-m = 3406" = 56' 46" 


j-6.4573 

j-3.2287 
f-3.3188 


s+d+m = 3990" = 66' 30" 

s-d-m = 1426" = 23' 46" 
s-d+m = 2010" = 33' 30" 


+0.2478 
1^.769 = l h 46* 8*. ( 

Then I4 h 6 m 29 s = t 
l h 46 m 8 s 


[-9.9099 
)h.813 = 48" 

14 h 6 m 

4S m 


47 s . 

29» 

47 s 


12 h 20 m 21 s Beg. of part, eclipse. 
15 h 52 m 37 s End of part, eclipse. 


13»» 17" 
I4 h 55 m 


42 s Beg. of total eclipse. 
16 s End of total eclipse. 



* At the same time that t is taken from the tables by means of its tangent, take 



Gr. M. Time. 


Long, of N. H. 


N. H. Mean Time 


11 24 8 


4 51 46 


6 32 22 


12 20 21 


" 


7 28 35 


13 17 42 


" 


8 25 56 


14 6 29 


<< 


9 14 43 


14 55 16 


" 


10 3 30 


15 52 37 


« 


11 51 


16 48 50 


<< 


11 57 4 



ECLIPSES OF THE MOON. 85 

And, proceeding in the same way with s'+d, we have 

llh 24"» 8» First contact with Penumbra. 
16 h 48m50s Last " " " 

Collecting these results into a tabular form, 

First contact with Penumbra. 
First contact with dark Shadow. 
First total Immersion in dark Shadow. 
Middle of Eclipse. 

Last total Immersion in dark Shadow. 
Last contact with dark Shadow. 
Last contact with Penumbra. 

T3TT o_J_/7 rn 

Magnitude of the eclipse* (= p^, see fig.) = --3 = 

3406" . . ,. . 

— — - = 1.720 on the southern limb. 
1 you 

88. To project a lunar eclipse. 

The same results may be obtained by means of an easy pro- 
jection, with the advantage of having, as it were, an exact pic- 
ture or representation of the particular features of the eclipse. 

Draw the axes AD and CF (fig. 20) perpendicular to each other ; 
and place the plus sign above, and the minus below, on the line 
AD, — and on the line CF, at the left and right respectively. Set 
off the values of p (Art. 87.) from S upon the line CF, in direc- 
tions corresponding to their signs ; thus —2005" will be SP to- 
ward the right hand, and +1821" will be SP' toward the left. 
In the same way make P XIII, . . . P' XV, equal to +1183" . . . 
—467" respectively. Through the points XIII, XIV, XV, draw 

out its logarithmic secant, cosine, and sine. This will save the trouble of a second, 
third, and fourth reference to the same place. 

* The magnitude of an eclipse, in a partial eclipse, is the greatest distance to 
which the shadow advances on the moon, measured in parts of her diameter. In a 
total eclipse, the shadow being supposed to have advanced, not only upon the 
moon, but beyond it on the other side, the " magnitude" is still measured from the 
point of the moon's limb, nearest the centre of the shadow at the middle of the 
eclipse, across the moon's disc to the edge of the shadow beyond, and is therefore 

always equal to — i~ — . 

The magnitude of the eclipse is frequently given in digits, or 12ths of the moon's di- 
ameter ; thus in the present case, 12x1-720, or 20.6 digits are eclipsed on the moon's 
southern limb. 

The eclipse is said to be on that limb of the moon, which is nearest the centre of 
the shadow at the middle of the eclipse, and is therefore on the northern or southern 
limb, according as q' is negative or positive. 



86 OCCULTATIONS AND ECLIPSES OF THE SUN. 

BE, the relative orbit of the moon, and through S, RSU at right 
angles to the same, cutting it in M. 

With the semidiameter of the earth's shadow (s) and the cen- 
tre S, describe the circle ACDF, and with the same centre, and 
the radii s+d and s—d, cut the relative orbit in the points B, E, 
and B', E'. With the radius d, or the moon's semidiameter, and 
centres B, B', M, E', E, describe circles, representing the moon, 
at the moments of the different phases. Then, by ascertaining 
the values of XIII XIV, or XIV XV on a scale of equal parts, 
the distances XII B, XIII B', XIV M, XIV E', and XV E, will 
show in parts of the value of an hour, how long after 12 h , 13 h , 
&c, respectively, the different phases take place. 

RTJ 

By a scale of equal parts, ~~ will be the magnitude of the 

eclipse. 

To determine what point of the moon's limb, reckoning from 
the north point v around to the right hand, the shadow will first 
touch, measure the arc A<p, and add 180° ; this will equal the arc 
vxp. So for the last contact, the arc AD(p'+180° ==v'<p'. These 
two angles are for the present eclipse, 

Ang. from N. point. 

For First Contact with dark Shadow, 241° 8' 

For Last Contact with dark Shadow, 71° 12' 

If carefully projected on a scale three or four times larger than 
that of the figure, the particulars of a lunar eclipse may be ascer- 
tained with sufficient exactness, since on account of the great 
indefiniteness of the edge of the moon's shadow, they are neces- 
sarily to some degree uncertain. 



CHAPTER VII. 

OF OCCULTATIONS AND ECLIPSES OF THE SUN. 

89. The moon, in its monthly revolution, often passes over 
and hides from our view those stars which lie in and near the 
ecliptic. The planets also, because, like the moon, their orbits 



OCCULTATIONS AND ECLIPSES OF THE SUN. 87 

are but little inclined to the ecliptic, are occasionally covered in 
the same manner, but less frequently than the stars, in proportion 
as they are fewer. The only other body which the moon can 
thus occult is the sun, whose annual path around the sun is the 
ecliptic itself. The two former phenomena are called occulta- 
tions of a star or planet, the latter an occultation or eclipse of 
the sun. 

Let us regard, for a moment, the zodiac as a wide path around 
the heavens, in which stars of unequal magnitudes are scattered 
here and there, and in which the sun, planets, and moon are 
moving, but the moon nearer and swifter than any of the rest. 
Neither the moon or planets ever retrace the same exact line 
along this path, but in repeated revolutions mark out an endless 
variety of courses, all described within the zodiac. This is the 
reason why no two eclipses or occultations are ever exactly 
alike, for the moon never overtakes any one body on precisely 
the same course in successive revolutions, and the path of con- 
course is so wide that she much more frequently passes by it to 
the one side or the other. 

It is evident from this illustration, that occultations, whether 
of the stars, the planets, or the sun, are all phenomena of the 
same class. The only difference between them is this, that the 
stars have neither sensible diameter, motion, nor parallax, while 
the sun and planets have all these. The calculation of an occul- 
tation, when a star is the body occulted, is of course much the 
simplest, and to this case, then, we will first give our attention ; 
and since the same general rules are applicable to all bodies, it 
will not be difficult afterward to modify our formulae, so that 
they shall include the sun and planets. 

90. The parallax of the moon. 

The lunar parallax is that which renders tne calculation of an 
occultation of the sun or a star more difficult than that of an 
eclipse of the moon. Any considerable change of place on the 
earth's surface always shifts the moon's apparent place in the 
heavens in an opposite direction, on account of the proximity of 
that body. If all the inhabitants in that hemisphere of the earth 
turned towards the moon were looking towards her at the same 
time, the different points to which they would refer her in the 



88 OCCULT ATIONS AND ECLIPSES OF THE SUN. 

heavens would be scattered over a circle about 2° in diameter. 
Suppose the sun or a star were at that moment situated within 
this circle ; there would necessarily be an eclipse somewhere on 
the earth. But as the sun is only £° in diameter, it would de- 
pend entirely on the place of the observer whether the moon 
were apparently so situated within this circle of 2° as to cover 
it, either partially or totally, or to be wholly separated from the 
sun's limb. Thus in the last eclipse of Sept. 1838, an observer 
at Washington would see the moon almost exactly on the centre 
of the sun ; north of Washington, at Boston, it would only ob- 
scure his southern limb, while at Charleston, South Carolina, it 
would cover only his northern limb ; still farther south, as at Rio 
Janeiro, South America, the moon would be apparently thrown 
wholly off of the sun to the north. So according as the observer 
was west or east of Washington, it would eclipse the sun earlier 
or later, because apparently thrown forward or backward in its 
path. 

91. Statement of the principles of calculation of a solar eclipse. 

In a lunar eclipse, it will be recollected that the places of the 
earth's shadow and moon in right ascension and declination be- 
ing given for several instants of time, and the differences be- 
tween these being represented by p and q, the times of first and 
last contact were those at which 

s and d being the semidiameters of the moon and shadow. 

Let us transfer our ideas to the small space in the heavens oc- 
cupied by the bodies in a solar eclipse or occultation, making the 
sun correspond to the earth's shadow represented in fig. 20. 
The case would be precisely the same as in a lunar eclipse, if the 
moon had no parallax ; and then at the moments of contact, 

IpZ+qSf =D+d, 
D and d being the semidiameters of the sun and moon. 

And the case will be precisely the same as in a lunar eclipse, 
if we calculate the amount of the moon's parallaxes or displace- 
ments in right ascension and declination for each of the given 
instants, and applying them to her true places, obtain her appa- 
rent places. If, in our supposed circle of 2°, the limit of the 



OCCULTATIONS AND ECLIPSES OF THE SUN. 89 

moon's possible displacements, we make —u and — v equal to the 
displacements in R.A. and Decl. as seen by us, the times of the 
contacts would be when 

\{p-uy+{q-vff =D+d. 
So that a solar eclipse only differs from a lunar in calculating 
the displacements or parallaxes of the moon in right ascension 
and declination, and so using the differences between her appa- 
rent places and those of the sun as the sides of a right angled 
triangle, of which the sum of their semidiameters is the hypothe- 
nuse. (See Art. 85.) 

92. Parallaxes of the moon in right ascension and declination. 

We wish familiarly to illustrate to the student the effect which 
the vertical parallax or depression of the moon has in changing 
its right ascension and declination, at different hours of the day. 
Clear ideas on these points will aid him much in understanding 
the more intricate formulae we shall presently have occasion to 
employ. The learner will see, by reference to fig. 21, without 
a minute explanation of its parts, that if HO represent the 
southern horizon, AB the meridian, and IX III XXI the moon's 
diurnal path through the southern sky from her rising to her set- 
ting, the moon will be depressed in the vertical circles by the 
spaces IX 9, VII 7, IV 4, III 3, &c, and less and less as she rises 
in proportion to the sine of her zenith distance, (Olmsted's Astron. 
Art. 83.) Her apparent path will be 9 7 4 3 M' 21, every where 
below her true one IX VII IV III M XXI. Now, while she is 
rising from the left hand toward the right, on the east side of the 
meridian, the horary circles, or those of right ascension, being per- 
pendicular to her apparent path, as PP', QQ', must slope down- 
wards and to the right hand. The moon by her vertical de- 
pression will therefore be thrown forward in right ascension by 
the spaces C7, D4, but less and less as she approaches the meri- 
dian, where the parallax in right ascension is nothing at all. On 
the west side of the meridian, on the other hand, we see why, 
for a similar reason, the moon's right ascension is diminished by 
her parallax, and she is thrown backward in her course by the 
spaces El, FM', G 21. 

The moon's parallaxes in declination are the same as the 

12 



90 OCCULTATIGNS AND ECLIPSES OF THE SUN. 

breadths of the space between her true and apparent diurnal 
paths, and, unlike the parallaxes in R.A., vary but little through- 
out her course both above and beneath the horizon. The reason 
is, that while the parallaxes in declination III 3, IE, MF, obvi- 
ously bear a smaller proportion to their respective vertical par- 
allaxes as the moon goes down, the vertical parallaxes themselves 
grow larger, and thus compensate the effect. In fact, the par- 
allaxes in declination increase slightly from the meridian to the 
horizon in this latitude. 

We see, that the parallaxes in right ascension and declination 
form the two sides of a small right angled triangle of which the 
vertical parallax is the hypothenuse. 

From this slight illustration of the parallaxes, we may already 
draw some obvious and useful conclusions. When the moon is 
at the east of the meridian, or rising, she is thrown forward in 
her course, and consequently an eclipse or occultation occurs 
earlier than the time of true conjunction in right ascension. 
And since the eclipse happens earlier in its ascent towards the 
meridian than it otherwise would have done, it is evident that it 
must happen also at a greater distance from the meridian. So, 
on the other hand, when the moon is west and declining, an oc- 
cultation always occurs later than the conjunction in R.A. ; yet, 
as before, at a greater distance from the meridian. 

93. Reduction to the sphere. 

The parallax of the moon is still farther modified by the sphe- 
roidal figure of the earth. The sine of the horizontal parallax 
of the moon is equal to the earth's radius divided by the moon's 
distance from the earth's centre. But in an oblate spheroid like 
the earth, the radius is not the same in different latitudes, but 
decreases from the equator to the pole ; the sine of the horizontal 
parallax must, of course, decrease in the same ratio. Table VI. 
contains the corrections for every degree of latitude, to be sub- 
tracted from the horizontal equatorial parallax, as given in the 
Nautical Almanac for every 12 hours. 

But the spheroidal figure of the earth affects parallax in still 
another way. The zenith from which we usually reckon zenith 
distances, is that point where a plumb-line, or a perpendicular to 
the surface of still water, if produced, would cut the sphere of 



OCCULT ATIONS AND ECLIPSES OF THE SUN. 91 

the heavens. Let PSQ (fig. 22.) represent the earth, P and Q 
being the compressed poles, S the place of the spectator, and 
ZSN a perpendicular to the surface of the spheroid at that point. 
The zenith to which parallaxes are referred, is the point where 
the radius OS if produced would cut the celestial sphere, because 
parallaxes are always measured from the moon's true place as 
seen at the centre of the earth. A reduction is therefore neces- 
sary of the true zenith Z to the reduced zenith Z'. The reduced 
zenith is always nearer to the equator than the true zenith, be- 
cause in an ellipse, the angle SOP is always greater than SNP ; 
the reduced latitude is consequently always less than the true lati- 
tude by the angle ZSZ'. 

Table VI. contains the values of ZSZ' for every degree of 
latitude ; by simply subtracting the angle found in the table from 
the known latitude of a place, we obtain its reduced latitude. The 
parallaxes of the moon, calculated with this latitude, will be such 
in direction and amount as if reckoned from the reduced zenith. 

Referring to fig. 22, it is easily seen, that by employing the re- 
duced radius OS, and the reduced zenith Z', or what is the same 
thing, the reduced latitude SOE, we render the parallaxes such 
as they would be in an imaginary sphere described around the 
centre O with the radius OS. Let P'SQ' be such a sphere for 
the place S ; Z' would be the true zenith of S on such a sphere, 
and SOE its true latitude. Such an ideal sphere might be de- 
scribed for every different place on the earth's surface, and the 
reduction of parallaxes for any particular place is hence termed 
the Reduction to the Sphere. 

94. Augmentation of the ?noon , s semidiameter. 

The immersion or emersion of a star takes place when its dis- 
tance from the moon's centre is equal to her semidiameter. But 
this semidiameter is different for different places on the earth's 
surface. The earth's centre, as before, is the standard of refer- 
ence. For this point the moon's true semidiameter MOD (fig. 
22.) is calculated in the Nautical Almanac for every 12 hours. 
But the apparent semidiameter MSC for any point S is always 
larger than this when the moon is above the horizon, because 

MS 

MS is then less than MO. Let A = ^y or the proportion of 



92 OCCULTATIONS AND ECLIPSES OF THE SUN. 

the moon's apparent distance to her true distance MO taken as 
unity, and let d and d' be the true and apparent semidiameters of 
the moon ; then OM being unity or radius, sin d = MD ; also 
A sin d' = MC .*. sin d = A sin d\ since MC and MD are radii of 
the same sphere. The true and apparent semidiameter of the 
sun are exactly the same in all calculations, the difference be- 
tween them being imperceptible on account of his great distance. 
The semidiameter of the moon is always so small, that no er- 
ror of moment is introduced by supposing sin d = d, and sin d'=d'. 
And the same may be said of the parallaxes ; — for instance sin 
< = * nearly. 

95. Investigation of the formulce for calculating an occultation. 

At this stage, some previous knowledge of Spherical Trigo- 
nometry will be found of much advantage, and in order to re- 
lieve the memory, and render reference to other text-books need- 
less, one or two equations in common use are here inserted. 

If the three angles of any spherical triangle be denoted by A, 
B, and C, and the three sides by #, b, and c, we have these equa- 
tions : 

In a triangle right angled at C, 
Formula 1. sin a = sin c sin A. 

In an oblique angled triangle, 

Formula 2. cos a = cos b cos c + sin b sin c cos A. 

All the known methods employed by astronomers, in investi- 
gating rigorously the theory of occultations, presume in the 
reader a wider range of mathematical acquirement than is usu- 
ally embraced in the course of most American colleges, and are 
too tedious and intricate to be introduced into the body of this 
work. It is therefore thought preferable to adopt a formula, 
which has been demonstrated analytically by Prof. Bessel of 
Germany,* and which, it is believed, possesses some advantages 
over every other ; and assuming it as true, to give such a de- 
monstration or explanation of it by geometrical illustration, as 
shall render its several parts clear and intelligible to the general 
student. Yet, for the benefit of those desirous of a more tho- 

* Prof. Bessel's papers on this subject were originally published in Schumacher's 
Astronomische Nachrichten, and are translated into English in the 6th and 8th vol- 
times of the Philosophical Magazine, New Series, 1829 and 1830. 



OCCULTATIONS AND ECLIPSES OF THE SUN. 93 

rough investigation, we have thrown into the form of a note at 
the end of the work a demonstration, which, although rigorously 
exact, will be found to require no previous knowledge of Alge- 
braical Geometry, nor any with Spherical Trigonometry, except 
of the few common formulae there assumed as known. 

96. In fig. 23, let EQX be the equator, and P its pole ; and 
let E be the vernal equinox from which right ascensions are 
reckoned in the direction EQX. Let M be the true place of the 
moon, and S the star to be occulted ; and PMR and PSQ hour 
circles passing through them. Let Z be the reduced zenith, and 
PZT the meridian of the place of observation. 

Also, make QX == 90°, and join PX ; then PQX is a triangle, 
having each side and each angle = 90°, and QX, PQ, PX are the 
arcs of great circles, whose poles are P, X, and Q. 

Produce QP to Y, making PY = SQ, and join XY ; then YSX 
is a triangle of the same kind with PQX. From X and Y draw 
arcs of great circles through Z and M. 
Let EQ = A = apparent R. A. ) 
QS (= P Y) ==4;= - Decl. S of the hod ? occulted - 
ER = a = true R. A. 
RM =6 = " Decl. 

* = horizontal parallax 
d = horizontal semidiameter 
ET =fj, = sidereal time ) of the place of observa- 
TZ = 9 = reduced latitude S tion. 

Then QR = angle BPM = a-A 
QT= angle CPZ = ^-A 
The quantity it denotes the parallax of the moon reduced for 
the latitude of the place by Table VI. 

We have then at the time of immersion or emersion of a star, 
(I.) sin 2 d— jcos 5 sin (a — A) — sin <g cos 9 sin (jx— A)l* 
+ jsin 6 cos d— cos 6 sin J cos (a — A) 

— sin it Jsin 9 cos A— cos 9 sin A cos (p—A)\ j 2 . 
This is, in effect, the equation demonstrated by Prof. Bessel in 
the Phil. Mag. for 1829, vol. VI. p. 338 ; slightly altered in no- 
tation, and cleared of fractional expressions. We shall give an 
approximate demonstration of this formula by a geometrical 
construction. 



of the moon. 



94 OCCULTATIONS AND ECLIPSES OF THE SUN. 

If we take the several parts of this equation, and refer them 
to the figure, we shall find 

1. cos 8 sin (a— A) =cos RM sin BPM = sin PM sin BPM = 

(Form. 1.) sin BM = cos XM. 

2. cos <p sin (|x — A) = (for the same reason) cos XZ. 

3. sin 8 cos J— cos 8 sin A cos (a— A) = sin RM cos 

QS-cos RM sin QS cos BPM = cos PM cos 

PY+sin PM sin PY cos MPY* = (Form. 2.) cos YM. 

4. sin 9 cos d — cos 9 sin 4 cos (ja— A) = (for the same 

reason) cos YZ. 

Therefore, if we let 

cos XM =p = cos 8 sin (a— A). 

cos YM =q = sin 8 cos 4— cos 8 sin 4 cos (a— ^4). 

sin ^ cos XZ = w = sin * cos 9 sin (p — A). 

sin if cos YZ = i; = sin if \ sin 9 cos ^— cos 9 sin d cos (|x— ^4) \. 

the equation above stated becomes 

(II.) ( p _ M )2 + fe-v) 2 = sin 2 d. 

If, then, two great circles cut one another at right angles at 

the point of the star to be occulted, one of them being an 

hour circle, p and q will be the cosines of the arcs joining the 

poles of these two circles with the true place of the moon ; and 

u v 
and - — will be the cosines of the arcs joining these two 

sin if sin tf jo 

poles with the reduced zenith of the place. 

97. The small portion BSAM of the heavens, in which the 
moon and star are situated, may be considered as a plane sur- 
face, and portions of great circles crossing it, as straight lines.f 
Let PS and XS (fig. 24.) be the two circles cutting one another 
at right angles at the place of the star S, of which PS is an 
hour circle ; let M be the true place of the moon, and M' its 
apparent place as depressed in a vertical circle. At the moment 
of immersion or emersion the distance between the star and the 
apparent place of the moon will be equal to her semidiameter, 

* The sign is here changed, because cos MPY= —cos BPM. (See Table I.) 
+ In fig. 23, we look on the celestial sphere from an imaginary point outside of it ; 
in figs. 24 and 25, from our real station within it. The small portion BSAM in fig. 
23 is therefore reversed in a horizontal direction in figs. 24 and 25. The latter re- 
present an occultation and eclipse occurring to the west of the meridian. 



OCCULTATIONS AND ECLIPSES OF THE SUN. 95 

or M'S. Draw MB and M'F perpendicular to PS, and AM and 
GM' parallel to the same. Then M'S 2 = (BM-GM) 2 +(BS- 
GM') 2 . This equation, it will be seen, is much like equation II. 
And since M'S = sin d or d, if we can show that BM, GM, BS, 
GM' are equal, or very nearly equal to p, u, q, and v respective- 
ly, we shall demonstrate equation II, and consequently, its more 
expanded form in equation I, sufficiently for all the purposes 
of illustration. 

(1.) p is equal to BM ; for (see fig. 23.) p — cos XM = sin 
BM (fig. 24.) = BM very nearly, since in an eclipse BM is always 
very small. 

(2.) q = BS ; since q = cos YM = sin AM = AM = BS. 

(3.) u = GM ; for (see figs. 23 and 24,) MM', or the parallax 
in altitude is equal (Olmsted's Astron. Art. 83,) to sin if x sin. 
app. zen. dist. ; but the app. zen. dist. of the moon differs from 
that of the star by a quantity less than her semidiameter ; there- 
fore MM' = sin if sin ZM' = sin if sin ZS very nearly. But 
GM=MM' sin GM'M = sin if sin ZS sin PSZ = (Form. 1.) 
sin if sin CZ (see fig. 23,) = sin if cos XZ. But u = sin if cos XZ ; 
therefore u = GM. 

(4.) In the same way it may be shown that v is equal to GM' ; 
since GM' =MM ; cos GM'M = MM' sin ZSX. 

It is easily seen in fig. 24, that the portions of equations I. 
represented by p and q, express the differences between the star 
and the true place of the moon measured on the circles XS and 
PS ; while the parts represented by the symbols u and v are 
equal to the parallaxes of the moon on the same circles. Thus 
we find our rough statement of the principles of calculation (see 
Art. 91.) confirmed. 

We have in this demonstration supposed the sines of very 
small arcs equal to the arcs themselves, and have neglected the 
minute augmentation of the moon's semidiameter due to her al- 
titude, (Art. 93.) But in the rigorous solution of the problem, 
(see Note at the end of the volume,) all these have been taken 
into account. It expresses the condition, that the moon, con- 
sidered as a sphere, should have for a tangent the ray proceed- 
ing from a star to the observer, at a given point on the earth, 
considered as a spheroid ; and as no limitation was introduced 
into the analysis, it may be regarded as mathematically accurate 



96 OCCULTATIONS AND ECLIPSES OF THE SUN. 

And equation II, being identical with equation I, has the same 

foundation for perfect correctness. 

98. Formula for a solar eclipse. 

To adapt the foregoing formula to eclipses of the sun, some 

modifications are necessary. If we take the square root of eq. 

II, it becomes 

i 
\{p — u) 2 -{-(q— vf\ 2 = sin d. 

or since, by Art. 93, d — sin d nearly, 

(III.) l(p-uY + ( q - v yii = d. 

The same strict analysis by which the equation for occupa- 
tions was obtained, when applied to eclipses of the sun, leads 
to an extremely intricate equation ; from which, however, by 
striking out the higher powers of very small arcs, whose omis- 
sion cannot introduce an error of more than one or two tenths 
of a second of space, another equation results of comparative 
simplicity, and of correctness amply sufficient for all purposes 
of calculation. It is as follows : 

(IV.) {(p-uY+iq—v) 2 ^ =I)+d-D sin « cos z. 

where D denotes the true semidiameter of the sun, 
z " the sun's true zenith distance, 
and -7T " the difference of the horizontal parallaxes of 
the moon and sun. 

It is to be remarked that d and D in equations III and IV, are 
to be expressed in parts of radius as unity, and not in seconds 
of space. 

99. It will be seen that this differs from the formula for occupa- 
tions in two points ; ]st, the term *, which enters as an element 
into the values of u and v, and is also contained in the last term 
of the equation, does not express, as before, the moon's horizontal 
parallax, but the difference between her own and the sun's ; and 
2nd, the distance between the two bodies at the moment of con- 
tact, or the sum of their semidiameters is diminished by the term 
D sin if cos z. 

To explain the reason of these differences, let us have recourse 
to fig. 25. Let M be the true place of the moon, and S that of 
the sun at the moment of first or last contact. The moon, as 



OCCULTATIONS AND ECLIPSES OF THE SUN. 97 

before, is depressed from M to M', in the direction of a vertical 
circle ; the sun also, being about 400 times as far off as the moon, 
is depressed T £o as much, or from S to S'. The distance S'M' 
then equals the sum of their semidiameters, and S'M' 2 = M'F 2 + 
S'F 2 , as before. But M'F-BG = MH-(MG-SL) ; and S'F = 
SH-(M'G — S'L). Now SH is the difference of declination be- 
tween the true places of the sun and moon, or q (Art. 97), while 
MH =p, or their difference of place on the circle perpendicu- 
lar to PS. Also M'G— S'L = the difference of parallaxes on 
an hour circle, and MG— SL on the perpendicular circle. 
Therefore that the equation may be of the form (D+^) 2 = 
(p—uf+^q-vf^ which corresponds to S'M' 2 = |MH-(MG — 
SL)S 2 -f ISH-(M'G-S'L); 2 ,— since p and ?are equivalent to MH 
and SH, u and v should correspond to MG— SL and M'G— S'L ; 
that is, to the differences of parallax on the circles PS and XS. 
And, because the triangles MGM' and SLS' are similar, and the 
sines of the zenith distances of M' and S' are nearly the same, 
therefore these differences have the same ratio to the whole par- 
allaxes of the moon on these circles, as the difference between 
the horizontal parallaxes has to the moon's whole horizontal par- 
allax. Consequently, that u and v may correspond to the differ- 
ences MG— SL and M'G— S'L, the quantity sin <, which enters 
as a factor into their value, must refer to the difference between 
the horizontal parallaxes of the sun and moon. 

Taking the square root of the equation above \(p— iif-\- 

(q — v) 2 \^ =~D-\-d. The addition of the term — D sin «r cos z in 
eq. IV, depends on the difference between the true and apparent 
semidiameters of the moon, (Art. 93.) A slight consideration 
of fig. 25 shows us, that the sun's true zenith distance (z) never 
differs much in an eclipse from the moon's apparent zenith dis- 
tance ; z is therefore nearly equal to Z'SM, (fig. 22,) or the 
moon's app. zen. dist. We have then by Art. 93, 

d! : d : : MO : MS : : sin OSM (sin z) : sin MOZ'. 
Let OMS - n ; then since MOZ' - MSZ'-OMS, /. sin MOZ' = 
sin (z— n) — (Day's Trig. Anal. 208. II.) sin z cos n — cos z sin n 
= sin z — sin * sin z cos z, since cos n = 1, very nearly, and sin n 
= sin ie sin z, (Olmsted's Astron. Art. 83.) Then by the pro- 
portion above, we have 

13 



98 OCCULTATIONS AND ECLIPSES OF THE SUN. 

d sin z — sin it sin z cos z 

— = — . : = 1 — sin It COS z. 

a sin z 

Hence we see, that the semidiameter of the moon is always ap- 
parently augmented when above the horizon nearly in the pro- 
portion of 1 — sin it cos z : 1. 

Now it is manifest, that the distance between the moon's ap- 
parent place and the star, at the moment we see it occulted, is 
equal to her apparent semidiameter. But in eq. Ill, her true 
semid. d was used. In fact, the analysis by which that equation 
was obtained eliminates the quantity d' in the process, and in- 
troduces d ; and, (since the other parts of the equation are pro- 
portionably diminished,) this change might be expressed in fig. 
24 by supposing the scale on which it was projected to have been 
reduced in the proportion d' : d. When, therefore, we introduce 
the new quantity D (which represents either the sun's true or 
apparent semidiameter, Art. 94,) into eq. IV, or fig. 25, the same 
analysis requires that this also suffer first a similar reduction in 

the ratio -j f , and the quantity M'S' in fig. 25 becomes equal to 

d+D j, =d-\-J) (1— sin it cos z) = J+D— D sin it cos z, which is 

exactly the second member of eq. IV. 

The fact may be thus expressed, that, to correspond with the 
substitution of the true for the augmented semidiameter of the 
moon, the sun's semidiameter is diminished in the same ratio. 

100. The value of q in Art. 96 may easily be brought into a 
form better adapted for calculation. We there obtained, 
sin 5 cos A — cos 8 sin A cos (a — A) = q. 

2 siu 2 i (a- A) = 1 -cos (a— A) . . . (Day's Trig. Anal. Art. 210.) 
Transposing the 1, and multiplying by cos 8 sin A, we have 
— cos 8 sin 4-\-2 cos 8 sin A sin 2 \ (a— A) = — cos 5 sin d cos (a— A). 
Substituting this value of —cos S sin 4 cos (a— A) in the first 
equation, 

sin 8 cos J— cos 8 sin ^+2 cos 8 sin 4 sin 2 i (a— A) =q. 
But (Day's Anal. Trig. Art. 208, II.) 

sin 8 cos A— cos 8 sin A = sin (8—4) 
.*. sin (8—d)-\-2 cos 8 sin 4 sin 2 i (<x—A) =q. 
In formula IV, we have a quantity cos z, which cannot be ta- 



OCCULTATIONS AND ECLIPSES OF THE SUN. 99 

ken directly from the ephemeris, and* it must therefore be brought 
into a different form. For this purpose we have 

cos z = cos ZS (fig. 23.) = (Form. 2) cos PZ cos PS + sin PZ 
sin PS cos SPZ = sin TZ sin QS + cos TZ cos QS cos SPZ 
= sin (p sin ^+cos <p CO s A cos (p.— A). (G.) 

All our formulae are thus far expressed in parts of the radius 
of a sphere, whereas it is more convenient that they should be 
expressed in seconds of arc, of which 206264". 8 = radius. We 
will therefore change them so as to answer this purpose. 

p = sin (a— A) cos 8. 
Since sin (a. — A) is very small, we may substitute the arc, and 

(C.) p = (a— A) cos 8 ; . . . where a— A, are expressed in sec- 
onds instead of parts of the radius. 

Again, . . . q = sin (5—A)+2 cos 8 sin d sin 2 \ (a— A). 

8— J 
Here sin (8 — 4) and sin ? (a— A) are nearly equal to — -r and 

— — ; — -, and therefore 
rad. 

8-d , t . (a— Af 

+ 2 cos 8 sin J v 



* rad. 4 rad. 2 ' 

Multiplying, as before, by radius, to reduce from parts of ra- 
dius to seconds, 

(D.) q = 8 — 4+— cos <Ssin^(a— Af \ . . . where 8—4, and 
a— A, are expressed in seconds of arc, and r = 206265". 

In the expressions for u and v, sin it = — -r- nearly, and multi- 
plying by rad., 

(E.) u = <ie cos <p sin (ja— -4). 

(F.) v =ir J sin 9 cos A — cos <p sin ^ cos (p—A) j. 

To correspond with these changes, let d and D, in equations 

III and IV, instead of being expressed in parts of radius, be 

converted into seconds of space, and we shall have 

i 
(A.) \ (p — u) 2 -\-(q— v) 2 p =d, . . . for occultations. 

(B.) \(p— uf-\-{q— «) 2 }*=D+d— D sin* cos z,. . . for eclipses. 

The advantage of this change in the formulae consists in this, 
that the quantities d, D, *, &c, as taken from the Naut. Aim., 
are expressed in minutes and seconds of arc, and not in parts of 
the radius. 



100 OCCULTATIONS AND ECLIPSES OF THE SUN. 

The quantities a— A, and ft — A, must be converted from hours, 
minutes, and seconds, into degrees, minutes, and seconds, by 
multiplying by 15; they will thus be of the same denomination 
as the other quantities. 

101. In the above equations (A) and (B), if we consider that 
p, q, u, v, and cos z are but abbreviated symbols of their values 
in equations (C), (D), (E), (F) and (G), we shall see, that these 
our two principal equations are made up entirely of known quan- 
tities. Most of these quantities, however, are to be taken from 
the Naut. Aim., and are therefore known only, when the times 
for which they are to be computed are given. Now the times 
of beginning and end of occultation are the very quantities of 
which we are in search, and are, of course, unknown. We must 
therefore find them by some method of approximation, or of trial 
and error. 

Let T be the approximate time of greatest obscuration ; and 
let T-30 m , T, T+30 m , in the case of a star,— and T-l\ T, 
T+l h , in the case of the sun, be three instants chosen for the 
computation of the equation (A), or (B). It is evident, that these 
three instants will include, or nearly include, the whole occulta- 
tion or eclipse. 

But to find T, or the time of the middle of the eclipse, is a 
matter of some little difficulty. It is always nearly the same as 
the time of apparent conjunction in R.A. of the sun and moon. 
In fig. 21, suppose the sun at M' ; the moon, to be in app. conj. 
with it, must be apparently at M', but truly at M ; that is, she 
must have advanced in R.A. by the quantity HM, since the time 
of true conjunction. The time of app. conj. is therefore later 
west of the meridian, and earlier east of it, (see Art. 92, last 
paragraph,) than the time of true conj., by the time the moon 
occupies in describing HM. 

The Gr. m. time of true conj. is easily found, (see our exam- 
ple, Art. 104.) Suppose New Haven to be the place of obser- 
vation ; convert this Gr. m. time into N. H. sid. time, (by Prob. 
II, Art. 79,) and call it (*'. Also let the unknown N. H. sid. time 
of app. conj. be ft". Now HM = (see Art. 97. 3, and fig. 24.) 
u nearly = (Art. 99, eq. E,) ne cos <p sin (jx"— A), since HM is the 
parallax of the moon on the perpendicular circle at the time of 



OCCULT ATIONS AND ECLIPSES OF THE SUN. 101 

app. conj., or ^". Take from the Naut. Aim. the difference be- 
tween the moon's R.A. at two successive hours at the time of 
the eclipse, and that it may correspond with the quantity *, con- 
vert it from seconds of time to minutes of space by dividing by 
4 ; and call it h. Since the moon's change of R.A. in 1 hour 
equals h, her actual motion at a declination S, and in a direction 
parallel to the equator, will be (Art. 63.) h cos 8 per hour, and 

„,, HM ir cos (d sin fa" — J) sec 5 

the time of describing HM = , ? = ■ \ . 

° II cos o k 

Then since the time of app. conj. differs from that of true conj. 

by the time of describing HM, we have 

/v \ // , . tf cos <p sec <5 

(Y.) fx"=fx / + j- sin (y."—A)* 

On the second side of this equation, we find p", which being 
the object of our search, is of course unknown, and requires 
that the equation, being of a transcendental nature, should be 
solved by successive approximations, as is done in our example, 
Art. 104. 

Having found ^", the N. H. sid. time of app. conj., convert it 
into the Gr. m. time of the same event. This will be nearly 
the middle time of occultation, or T ; and subtracting and add- 
ing 30 m or l h , as the case may be, we have the instants of time 
required. 

102. Now if we compute equation (A) or (B) for the three in- 
stants thus arbitrarily chosen, we shall find that at none of them 
are the two members of the equation (which we will represent 
by m and n respectively) equal to one another ; for such equality 
takes place only at the two unknown instants of first and last 
contact. The second member, n, being the sum of the semidi- 
ameters of the two bodies, remains nearly constant during an 
eclipse ; but the first member, m, which expresses the apparent 
distance between their centres at the instants for which it is 
computed, is of very rapid change. From a value greatly above 
that of n before the eclipse, it gradually diminishes, becomes 
equal to n at the moment of first contact, and still decreasing, 

* Here sin (ji"—K) becomes negative at the east of the meridian, (because p"— A is 
in that case between 12 h and 24 h , or between — h and — 12 b ,) and therefore changes 
of itself the sign of the fraction, so as not to require the double sign after n'. 



102 OCCULT ATIONS AND ECLIPSES OF THE SUN. 

reaches a minimum at the time of greatest obscuration ; thence 
it increases, and again becomes equal to n at the moment of last 
contact. By considering these circumstances, and noticing 
whether at T-30 m and T+30 m , (or at T-l h and T-fl h ,) m is 
greater or less than n, and how much, we can form some 
idea where the times of first and last contacts in an occultation 
or an eclipse occur amidst our three assumed instants, and inter- 
polate for them accordingly. 

The quantity m is so variable in its rate of change, that it is 
unsafe to interpolate from three values for any intermediate ones ; 
but p— u, and q — v, which form the sides of a right angled tri- 
angle, of which m is the hypothenuse, are comparatively uniform 
in their change. Therefore, from the three values of p — u, of 
q — v, and of n, which have been computed, interpolate interme- 
diate values at intervals of 5, 10, or 15 minutes, to such an ex- 
tent as will probably include the two required unknown times. 
For example, if at T— l h , m is a little greater than n, interpolate 
for T-50 m and T-40 m . Then if at T-50 m , the excess of m 
over n =a, and at T — 40 m , m is less than n by b, say 

Time of first contact = T-50 m +10 ra -^r. 

a+b 

If the time of first contact, as thus obtained, is found to be 
about T—43| m , to obtain a somewhat more accurate result, in- 
terpolate again for T— 44 m , and T— 43 m , and let the excess and 
defect of m at these instants be a' and V respectively ; then we 
have for a second approximation, 

a' 
Time of first contact = T— 44 m +l m — -=-. 

a'+b' 

And this process, in all ordinary cases, will give the times of 

first and last contact with sufficient accuracy. 

103. Synopsis of formula for stellar and solar occultations. 

We collect our formulae into one general view, for the sake of 
more convenient reference during calculation. 

First, in an occultation, at the moment of immersion or emer- 
sion, 

(A.) {(p-uf+iq-vY^d. 

Secondly, in an eclipse of the sun or a planet, at the moment 
of first and last contact, 



CALCULATIONS BELONGING TO OCCULTATIONS, ETC. 



103 



(B.) \(p-u) 2 +(q-vYf = T>+d-T) sin « cos z. 
The auxiliary quantities are expressed by the following equa- 
tions : 

p = {ol—A) cos 5. 

1 



(C.) 
(D.) 



q=8. 



■4+ — cos S sin 4 (a— A) 2 . 
2r 



(E.) u ==# cos 9 sin (^—A). 

(F.) v = * j sin 9 cos 4— cos 9 sin d cos (p—A) \. 

(G.) cos z = sin 9 sin ^ + cos 9 cos A cos (p— A). 

In these equations, ie always denotes the difference between 
the horizontal parallaxes of the moon and of the body occulted 
at the place of observation. 

Finally, to find T, or the middle time for which the equations 
are to be computed, we have 



(Y.) 



. , if cos 9 sec 5 . 
F+ 7 sin (^' 



■A). 



Of the calculations belonging to Stellar and Solar Occultations. 



104. Example. Let it be required to find the times of first and 
last contact of the moon and sun in the annular eclipse of Sept. 
18, 1838. 

From the tables of the Nautical Almanac for 1838, we require 
the following data : 



Mean Time. 



1838. 
Sept. 
17U oh 
I8«i " 
19 d « 

20J " 



Sun's R.A.=A. Sun's Decl.=A. 



llh.38™25 s .08 
" 42 0.G1 



36.16 
11.75 



4-2° 20' 14".3 
+1 56 58 .2 
+1 33 39 .5 
-j-1 10 18 .6 



Sun's semid.=D. 
15' 56".7 
" 57 .0 
" 57 .3 
" 57 .5 



Sid. Time of 

Gr. Mean Noon 

lib 47« 50U8 



From p. II. of 
the month. 



Sept. 


Moon's se- 
mid. = d. 


Moons's eq. 
hor. par. 


17«i 12b 
18 

18 12 

19 


14' 42".3 
" 41 .5 
" 41 .1 
" 40 .9 


53' 57".7 
" 54 .8 
" 53 .3 
" 52 .8 



From p. III. of 
the month. 



Sept. 18^ 

7 h m 
8 h « 
9h « 
l h « 



Sept. 18 



Moon's R.A. = a 
llh 41m 335.74 
" 43 17 .79 
" 45 1 .79 
" 46 45 .75 
" 48 29 .67 



Moon's Decl. = i 
+20 57' 14".9 
-f-" 43 4 .6 
-f-" 28 53 .9 
+" 14 42 .6 
+" 31 .0 



From pages V-XII 
of the month. 



Sun's I 
hor. par. From p. 266 immediately 
8".5 after the months. 



104 



OCCULTATIONS AND ECLIPSES OF THE SUN. 



The true conj. of the sun and moon in R.A. will plainly be 
about 18 d 8 h , or a little before ; and if we find the sun's R.A. 
for 18 d 7 h and 18 d 8 h respectively, we have 



]8J7h 
18<i8h 



11M3™ 3*.49 llh 41™ 33S.74 
11 43 12.47U1 43 17.79 



Diff. between the two 



a-A 
- 89*.75 Then -95*.07 : -89*.75 : : 60™ : 56™ 38% 
-j- 5 .32 and 18^ 7 h 56™ 38* = Gr. mean time of 

true coni. in R.A. 

-95 .07 



Now to find the time of apparent conjunction by Art. 100, 
we have 



On Sept. 18, 7^ 56™ 38* Gr. m. t. 



l 9 h 45m 46s .5 Gr. sid. t. ; by Prob. II. 
4 51 46 Longitude of N. H. 



1' = 14 54 



N. H. sid. t. of true conj. 



For finding **" a very rough 
method will answer. Referring 
to eq. Y, Art. 102, 

Take J u'-A = 14h 54™— 1 1* 43™ = 3* 11™ = 
48°. 
0r=4l°.3 = Lat. of N. H. 
7T = 53'.9 
m = 11^ 45™ 1« - llh 43™ 17s = 

104s = 26'.0 
3 = 3o 
and y."— A = yJ— A, for a first approxima- 
tion. 

I. 7T + 1.732 

I. h («.<:.) + 8.585 
I. cos <j> . + 9.877 
I. sec 6 . -{-10.001 



I *S°Lt¥SJ +0.195 
I. sin O'-A) + 9.87 



+ 0.06 
Nat. num. — {— l h .15 



1^9™. 



Then 3 h 11™ -|- lh 9™ = 4 h 20™ = 65°, or 
the 1st approximation to the value of p."— A. 

For a second approximation, 



ir cos <p sec S 
h 
I. sin 65° 



+0.195 
+9.957 



+0.152 
Nat. num. +lh.42 — l'* 25™. 
Then 3*» ll™+lh 25™ = 4^ 36™ = 69°, or 
the 2d approximation to the value of ji"— A. 

A third trial would make 
p"— A = 4 h 39 m ; but the second 
is always sufficient. 

Then /*"-A being 4 h 39™, 
u" — 16h 22™ N. H. sid. t. of app. conj. 
4 52 Long, of N. H. 



21 14 Gr. sid. time 
Gr. mean time of app. conj. 
T = 18d 9k 25™. 



9^ 25™ = 



As it is more convenient to have the times some aliquot part 
of an hour, call the middle time either 18 d 9 h 20 m , or 18 d 9 h 30 m , 
neither of which differs essentially from the middle of the 
eclipse. Assuming 9 h 30 m , for greater readiness in making pro- 
portions or interpolations, we have for the three times of calcula- 
tion, Sept. 18 d , 8 h 30 m , 9 h 30 m , and 10 h 30 m . # 



* Although these assumed times may be chosen almost at pleasure, and loosely in 
even minutes, or aliquot parts of an hour, yet once determined upon, they serve as 
instants of exact reference, and all computations dependent upon them must be 
conducted accordingly. 



OCCULTATIONS AND ECLIPSES OF THE SUN. 



105 



Corr. for Interp. 



24 h :+215 s .54 : 


gh 30m : +76*.34 


24h : -1398".7 : : 8h 30™ : -495".4+0".3 =- R' 15".l 




9 30 : -j-85 .32 


::9 30 : -553 .7-(-0 .3=- 9 13 .4 




10 30 : -f-94 .30 


::1030 : -611 .9+0 .3 =---10 11 .6 


For the sun's 
R.A.,by simple 






llh 4 2m 0.62 


For the sun's +1° 56' 58".2 


proportion* we 
have at 8'» 30 1 " . 




Decl., by inter- 


A = llh43"> 16*.96 


polation,we have at 8h 30"> . . . . A = +1° 48' 43". 1 


9 30 . 


" = " " 25.94 


9 30 " =+" 47 44 .8 


10 30 . 


" = " " 34.92 


10 30 ...." = +" 46 46 .6 



and so for a and 8, by simply halving the intervals between 8 h , 
9 h , 10 h , and ll h , we have 



At 8h 30™ 

9 30 

10 30 


a = llh 43^ 9*.79 

" = " 45 53 .77 
" = " 47 37 .71 


6 = 4-2° 35' 59" 
" = +" 31 48 
" = 4~" 7 36 


3 1 
.3 
.8 1 


a-A=-f- 52*.82 = + 792".4 
" =-f-147 .83 = +2217 .4 
" = -j-242 .77 = +3641 .5 


Again, 








Gr. sid. time. 


N. H. sid. time = p. 




p-A. 


At 8h 30™ 

9 30 

10 30 


201' 19m i3s.96 

20 19 23.82 

21 19 33.67 




15h 27m 27^.96 

16 27 37.82 

17 27 47.67 


3 h4 4 m 1K00 = 56<> 2' 45" 

4 44 11.88 = 71 2 58 

5 44 12.75 = 86 3 11 



For the moon's parallax, 



At 8h 30™ 

9 30 

10 30 



12h 



1".5 



Gh 30 s : 
9 30 : 
10 30 : 



I'M 
1".2 
1".3 



Moon's par. 
at 18 J Oh. 

53" 54".8 



Corr. Corr. for Sun'g 

for Spheroid, hor. 

Interp. Tab. VI. par. 

53'53".7-0".l - 4".5 - 8".5 = 

" 53.6-0".l - 4 .5 - 8 .5 = 

" 53.5-0".l- 4 .5- 8 .5 = 



53' 40".6 = 3220".6 
" 40 .5 = 3220 .5 
" 40 .4 = 3220 .4 

For all three times d = 14' 41".2, and D = 15' 57".l. 

(Lat.of N.H.) 41°18'28"-H'3" (corr. for spheroid, Tab. VI.) 41° V 25"= f 



Logs, of 


gh 30'" 


9'' 30'" 


10'' 30'" 


Logs, of 


8'' 30'" 


9 1 ' 30™ 


10'' 30"i 


cT-Af 


+2.89894 


+3.34584 


+3.56128 


sin <f> 




+9.81602 




sin (j/-A) 


+9.91881 


+9.97580 


+9.99897 


cos <p 


.... 


4-9.87697 




COS (/i-A) 


4-9^.74705 


+9.51155 


+8.83779 


IT 


+3.50794 


+3.50792 


4-3.50791 


cos & 


4-9.99955 


4-9.99963 


+9.99970 


sin 7T 


+8.19349 


4-8.19348 


+8.19346 


sin A 


+8.49990 


+8.49600 


4-8.49213 


D 




+2.98096 




cos A 


+ 9.99978 


+9.99979 


+9.99979 


1 
2r 




+4.385 





* When the second differences do not exceed 0-\04, or 0".4, interpolation will not 
add to accuracy, and simple proportion may be employed instead. 

t The logarithms of all the factors which enter into equations (B), (C), (D), (E), 
(F), and (G), arc here taken at once from the logarithmic tables, and arranged in 
order, so that during the progress of subsequent calculations, no logarithm need be 
looked out in the tables, but each may be copied at once from this short table in its 
order. 

14 



106 



OCCULTATIONS AND ECLIPSES OF THE SUN. 





8* 30m. 


9^ 30 1 ". 


10h 30™. 


Z. (a -A) 
Z. cos 8 


+ 2.89894 
+ 9.99955 


+ 3.34584 
+ 9.99963 


+ 3.56128 
+ 9.99970 


I. (a— A) cos 5 
P = 

Z. cos S 

I. sin S 

21. (a -A) 


+ 2.89849 
+ 79F.6 

+ 4.385 

+ 10.000 
+ 8.500 
+ 5.797 


+ 3.34547 
+ 2215".5 

+ 4.385 

+ 10.000 
+ 8.496 
+ 6.691 


+ 3.56098 
+ 3639".0 

+ 4.385 

+ 10.000 
+ 8.492 
+ 7.122 


=- cos <5 sin A (a— A) 2 

6 
A 


+ 8.682 
+ 0".0 

+ 2°35'59".3 
+ 1° 48' 43".l 


+ 9.572 
+ 0".4 

+ 2°21'48".3 
+ 1° 47' 44".8 


+ 9.999 
+ 1".0 

+ 2° 7'36".8 
+ lo 46' 46".6 


8-A 


+ 47' 16".2 

+ 2836".2 

+ 0".0 


+ 34' 3".5 
+ 2043".5 

+ 0".4 


+ 20'50".2 

+ 1250".2 

+ 1".0 


<1 = 

lit 

Z. cos (p 

I. sin (/u— A) 


+ 2836".2 

+ 3.50794 
+ 9.87697 
+ 9.91881 


+ 2043".9 

+ 3.50792 
+9.87697 
+ 9.97580 


+ 1251".2 

+ 3.50791 

+ 9.87697 
+ 9.99897 


u — 

lit 

I. sin <p 
I. cos A 


+ 3.30372 
+ 2012".4 

+ 3.50794 
+ 9.81802 
+9.99978 


+ 3.36069 

+ 2294".5 

+ 3.50792 
+ 9.81802 
+ 9.99979 


+ 3.38385 
4242G».2 

+ 3.50791 
+ 9.81802 
+ 9.99979 


it sin <p cos A 

lit 

Z. cos <p 
I sin A 
Z. cos (/i— A) 


+ 3.32574 
+ 2117".l 

+ 3.50794 
+ 9.87697 
+ 8.49996 
+ 9.74705 


+ 3.32573 
+ 2117''.0 

+ 3.50792 
+ 8.87697 
+ 8.49606 
+ 9.51155 


+ 3.32573 
+ 2117''.0 

+ 3.50791 
+ 9.87697 
+ 8.49213 

+ 8.83779 


7r cos sin A cos (//—A) 


+1.63192 

+ 42".8 


+ 1.39250 

+ 24".7 


+ 0.71480 

+ 5".2 


Z.D 

Z. sin 7r 
Z. sin (b 
I. sin A 


+ 2074".3 

+ 2.981 
+ 8.193 
+ 9.818 
+ 8.500 


+ 2092".3 

+ 2.981 
+ 8.193 
+ 9.818 
+ 8.496 


+ 211F.8 

+ 2.981 
+ 8.193 
+ 9.818 
+ 8.492 


Z.D 

Z. sin 7r 
Z. cos cp 

I. COS A 

Z. cos (/i — A) 


+ 9.492 
+ 0".3 

+ 2.981 
+ 8.193 
+ 9.877 
+ 10.000 
+ 9.747 


+ 9.488 
+ 0''.3 

+ 2.981 
+ 8.193 
+ 9.877 
+ 10.000 
+ 9.512 


+ 9.484 
+ 0".3 

+ 2.981 
+ 8.193 
+ 9.877 
+ 10.000 
+ 8.838 




+ 0.798 
+ 6".3 


+ 0.563 
+ 3".7 


+ 9.889 
+0".8 


D sin 7r cos z 


+ 6" .6 


+ 4".0 


+ l".l 



OCCULTATIONS AND ECLIPSES OF THE SUN. 



107 





8* 30«. 


9^ 30m. 


10^ 30">. 


D 

d 
— D sin 7r cos z 


4- 957". 1 

4- 88F.2 

-6".6 


4- 957". 1 

4- 881".2 

-4".0 


4- 957". 1 

4- 881".2 

-I'M 


D -j-d — T> sin * cos z 


4- 1831".7 


4- 1834".3 


4- 1837".2 


P 
u 


4- 791".6 
4- 2012".4 


4-2215".5 
-f-2294".5 


4- 3639".0 
4-2420".2 


p—u 

1 

V 


- 1220".8 

4-2836".2 
-^-2074".3 


-79".0 

4-2043".9 

4-2092".3 


4- 1218".8 

4- 1251".2 
4-2111".8 


q—v 


4- 761".9 


-48".4 


-860".6 



By rough trial : 



At 8'' 30^. 
(p-u)* = (1221")2 = (200 2 = 400 
(g-i>)» = ( 762" )2 = ( 13') 2 = 169 



1832" = 31': 



569 
</ 569 = 24' = m<n 



At 10h 30m. 
(p-u) 2 = (1219") 2 = (20') 2 = 400 
( ? -u) 2 = ( 861" ) 2 = (140 2 = 196 

596 
1837" = 31' = n. V 596 = 24' = m < n. 

Interpolating for spaces of 15 m before 8 h 30 m , and after 10 h 



30 r 





p—u. 


q-v. 


(p-u) 2 . (q-v)*. 


m 2 . 


m. 


n. 


m — n. 


8h 0m 

8 15 
8 30 

10* 30m 

10 45 

11 


-1733".2 
-1481 .9 
- 1220 .8 

4-1218".8 
4-1567 .6 
4-1926 .2 


4-1166".3 
4- 964 .2 
4- 761 .9 

- 860".6 
-1063 .9 
-1267 .4 


300398-.* 
219603 -. 

148546- . 
245742-. 


136026- . 
92968-. 

74063- . 
113188 -. 


436424-. 
312590- . 

222609 -. 
358930- . 


2089". 1 
1768 .0 

1492".0 
1894 .5 


1830".5 
1831 .1 

1837".2 
1838 .0 


4-258".6 
- 63 .1 


4-321 .7 

-345". 2 
4- 56 .5 


-401 .7 



Then +321".7 : 4-258".6 : : 15™ : 12m 4s. 
and — 401".7 : — 345".2 : : 15™ : 12m 53s. 



8 h 12 ra 4 8 . Gr. Time of first 
cont. at N. H. 
10 h 42 m 53 9 . Gr. Time of last 
cont. at N. H. 



We see at once that this eclipse is of extraordinary 

* The unit's place is omitted in these squares, because the logarithms do not fur. 
nish it readily, and it adds nothing to accuracy. 



108 



OCCULTATIONS AND ECLIPSES OF THE SUN. 



length, being no less than 2 h 30 m in duration at New Haven. 
The reason is, that not only is the true motion of the moon about 
at its minimum of velocity, but the effect of parallax is to make 
its apparent motion still slower. Instead therefore of finding 
the two times of contact a little more accurately by the simple 
repetition indicated in Art. 101, we shall, on account of the pe- 
culiar circumstances of the eclipse, employ a longer method, of 
needless accuracy in ordinary cases ; viz : — 

By proportion from the values of (jx— J) and 4 already known, 
find the same for 8 h 12 m , 8 h 13 m , and 10 h 42 m , 10 h 43 m ; and for 
these times recompute by the formulae, u, and that part of 
v which depends on cos (ft— J). Interpolate p, q, the first 

term of v, and n, for the same instants ; and finally recomposing 

i 
\(P~ #) 2 + (q~ v T\ 2 i ascertain at what two instants it is equal 
to n.* 





8h 12m. 


8* 13» - 


10»» 42m. l()h 43m. 


A 
p-A 


+ 1© 49' 0".6 
51° 32' 41" 


+ 1° 48' 59".6 
51° 47' 41" 


+ l°46'29".l 
89° 3' 14" 


+ 1° 46' 28".l 
89° 18' 14" 












Logs, of 


8h 12m. 


8h 13m. 


10h 42m. 


10h 43« 






sin A 
sin (/i-A) 
cos In— A) 

7T 


+ 8.50112 
+ 9.89381 
+ 9.79372 
+ 3.50795 


+ 8.50105 
+ 9.89531 
+ 9.79133 
+ 3.50795 


+ 8.49095 
+ 9.99994 
+ 8.21780 
+ 3.50791 


+ 8.49088 
+ 9.99997 
+ 8.08454 
+- 3.50791 





Calling the first member of v, v l9 and the 2nd, — v 2 , — then since 
Z. ie + Z. cos 9 +- Z. sin (^ — ^) — I. u, and Z. ir + L cos 9 + Z. sin 
J + I. cos (fx— J) =Z. v 2 , we have, by taking the logarithms from 
the above table, and the constant logarithms of sin 9 and cos 9 
as before, and adding them : 





8 h 12m. Qh 13m. 


10^ 42m. ioh 43m. 


Z. u 
u = 

I. v 2 

V2 
VI 


+ 3.27873 
+ 1899".9 

+ 1.67976 

+ 47".8 
+ 2117".l 


+ 3.28023 
+ 1906".5 

+ 1.67730 

+ 47".6 

+ 2117".l 


+ 3.38482 
+ 2425".6 

+ 0.09363 

+ 1".2 

+ 2117".0 


+ 3.33485 
+ 2425".8 

+ 9.96030 

+ 0".9 

+ 2117".0 


V=- 


+ 2069".3 


+ 2069".5 


+ 2115".8 


+ 2116".! 



* The common rule for interpolation regards only second differences ; and is not 
accurate, when, as in the present eclipse, the second differences are large, and values 
are interpolated for times much beyond the three instants chosen, on either side of 



OCCULTATIONS AND ECLIPSES OF THE SUN. 



109 





8»i 12m. 


8^ 13m. 


10h 42m. 


10!' 43m. 


p = 

p — u = 


4- 364".3 
— 1535".6 


+ 338".0 
— 1518".5 


+ 3923".6 
-f- 1493".0 


+ 3947".3 
-j- 1521".5 


q = 

q—v =. 


+ 3073".8 
-f- 1004".5 


+ 3060".6 
-j- 991".l 


4- 1092".6 
— 1023".2 


+ 1079".4 
— 1036".7 


n = 


1831".0 


1831".0 


1837".8 


1837".9 



Then, as before, 





p—u. 


q-v. 


(p-u)*. 


(q-v)K 


TO 2 . 


7/1. 


n. 


to— n. 


8t. 12'" 

8 13 

10h 42«. 

10 43 


-1535".6 
- 518 .5 

+1498".0 
-j-1521 .5 


+-1004".5 
-j- 991 .1 


235307 -. 
230534 -. 


100902- . 
98229-. 


336709 -. 
328813-. 


1835".0 
1813 .3 

1814".l 
1841 .1 


1831".0 
1831 .0 

1837".8 
1837 .9 


+ 4".0 
-17 .7 


-f21 .7 

-18" .7 

-f- 8 .2 


- 1023".2 
-1036 .7 


224399-. 
231494-. 


104694- . 
107475 -. 


329093 -. 
333969-. 


-26 .9 



Then +21".7 : -4- 4".0 : : lm : 1K06. 
and — 2C.9 : — 23".7 ::!»»; 52».96. 



8 h 12 m 11.1 = Gr. Time of first 
cont. at N. H. 
10 h 42 m 52 s .9 = Gr. Time of last 
cont. at N. H. 



8 h 12 m ir.l-4 h 51 ra 46 s = 3 h 20 n 25M*.. .Beginning of eclipse, 

N. H. mean time. 
10 h 42 m 52 8 .9-4 h 51 m 46 8 = 5 h 51 m 6 8 .9 ... End of eclipse, N. H. 

mean time. 



them. Thus the great length of this eclipse introduces an inaccuracy in these first 
results, not commonly to be apprehended. 

To remedy this defect, without repeating the calculation of all the formulae for 
times approximate to those of the contacts, let us examine whence the unequal va- 
riation of p—u and q—v arises. We at once see, that p and q, from their very na- 
ture, increase with tolerable regularity, while u and v, expressing the effects of par- 
allax, do not. The reason of this is, that u and v depend on (/* — A), which being a 
large arc, and of very different amount at the different times chosen, its sines and 
cosines do not vary at a nearly uniform rate, as the arc itself, and all the other fac- 
tors do. Therefore we need only recompute by the formulae the two short terms, 
which contain sin (// — A) and cos (// — a) ; and all the other terms may be interpola- 
ted from known values without appreciable error. 

* This result differs from that given for the same eclipse at New Haven in the 
American Almanac, 1838, — for two reasons. 1st, In that calculation the semidi^ 
ameter of the sun was assumed to be 5" greater than in the Naut. Aim. ; — and 2nd, 
The longitude of New Haven was taken at 4 h 51 m 51". Further, the latitude of N. 
H. in this work is assumed greater than that in the Amcr. Aim. by 30", causing a 
very trifling portion of the difference. The student, by increasing the numbers in 



110 



OCCULTATIONS AND ECLIPSES OF THE SUN. 



105. For the time of Greatest Obscuration. 

This usually differs only by a few minutes from the assumed 
middle time of eclipse, in the present instance 9 h 30 m . There- 
fore, interpolate for an interval of 5 m or 10 m both before and 
after 9 h 30 m , the values of p—u, and q— v, and combine as before 
to form m. The formulae for obtaining a minimum value (Prob- 
lem III, Art. 80,) will then afford an easy method of finding the 
time of " shortest distance of centres," which is the same as that 
of greatest obscuration. 



Time. 


p—u. 


q-v. 


m. 






9h 25™ 
9* 35 m 


- 180".0 

- 79".0 
+ 23".2 


+ 19".2 

- 48 '.4 

- 116".0 


+ 181".0 
+ 92».7 
+ 118".3 


- 88".3 
+ 25".6 


-l-113".9 



Then, Art. 80, 

Time of gr. obsc. = 9 h 25 ra + 1 x 5 ra : 
+ 113".9 +176".6 

' ~227^8 l ' 215 

Time of gr. obsc. = 9 h 31 m 22 s ; and by interpolation from the 
above three values for this time, m = +88".4, or shortest dis- 
tance. 

This process is sufficiently accurate for ordinary purposes, and 
even in this case, if the number of digits is all that is required. 
But on account of the rapid change of second differences, let us 
interpolate again for 9 h 30 m , 9 h 31 m , and 9 h 32 m , and our second 
result will be 

Time of greatest obsc. at N. H. =9 h 31 m 36 s , Gr. m. time. 

gh 31 m 36 8 _ 4 h 51 m 46 s = 4 h 39m 59s # # ^ JJ^ mean time of 

greatest obscuration. 

84".0 . . . Nearest approach of centres. This value is not in 
error by ,; .l. 



column n by 5", and repeating the proportion, may obtain results which differ from 
those of the Amer. Aim. only by an amount due to the difference of 5 s in the assumed 
longitudes. 

The longitude of New Haven by the eclipse of 1838, by subsequent occultations, 
and by late chronometric comparison with New York and Greenwich, is between 4 h 
51"» 46s and 4*> 51 m 47 s . 



OCCULT ATIONS AND ECLIPSES OF THE SUN. Ill 

106. For the times of Formation and Rupture of the ring in 
an annular eclipse ; and of Beginning and End of total darkness 
in a total eclipse. 

The equation for internal contact of the sun and moon dif- 
fers from that expressing their external contact only in the sign 
which connects their semidiameters. D (1 — sin * cos z) and d 
being, as before explained, the semidiameters of the sun and 
moon respectively, as proportioned to the other quantities of the 
formula, the equation of internal contact becomes 

\ {p-uf + (q-v) 2 ^ = D (1-sin * cos z) *d. 
When d > D (1 — sin <* cos z), the eclipse is total ; when 

d < D (1 — sin <tr cos z), an annular eclipse takes place. 

i 
It is farther evident, that when \ (p — u) 2 + (q— vf\^{=m), at 

its minimum value, is greater than D (1 — sin tc cos z) ^ d, neither 

annular nor total eclipse can take place ; when less, either one or 

the other must take place, according as D (1 — sin * cos z) is 

greater or less than d. 

In the present example, we have (D— D sin w cos z)^das 

follows : 

8 h 30 m 69".3 

9 h 30 m 71 // 9 /# w hence it equals 72".0 at 9 h 31 m 36 9 ; but m 
10 30 m 74 '.8 i s then at its minimum, and equal to 84".0. 
The eclipse is therefore neither annular nor total at New Ha- 
ven ; it is, however, very nearly annular, the moon, when near- 
est, overlying the sun's edge by only 12".0. Had the eclipse 
been annular, as the duration of the ring is always very short, 
the two times at which m — (D— D sin it cos z) ^d, could have 
been easily obtained from the values of these two quantities on 
this and the preceding page. 

107. For the points of first and last contact. 

Since the moon is altogether invisible before the first contact, 
it is desirable to know at what angle, reckoned from the vertex, 
or from the north point of the sun's limb, to the right hand 
around his circumference, the first indentation will be made upon 
his yet unbroken limb. For it is absolutely necessary to a good 
observation that the observer should know exactly at what point 
to look for the occurrence of an expected contact ; and he may 



112 



OCCULTATIONS AND ECLIPSES OF THE SUN. 



reckon from the north point, or from the vertex, according as he 
employs an equatorial with cross lines, or a plain telescope. 

In fig. 26, let mM. be the direction of the vertical circle pass- 
ing through the moon at the moment of first contact ; through 
O draw OZ parallel to it ; then Z will be the vertex of the sun, 
and N the north point, and the arcs ZNC and NC are those re- 

- (P~ u ) _ 



quired. But ^NOC = zlOMQ, and tan OMQ 



+ <*-*) 



-£— " And L ZON = L RmM ; and tan RmM = -. We have 
q—v v 

therefore the following formulae : 

u 

- = tan 0, where 6 is to be taken in the two first quad- 
rants, when u is positive. 

7) — U 

— =tan ^, where + is to be taken in the two first quad- 
rants, when p— u is negative. 
Then calling N and V the angles from the north point and 
vertex, 

In an occultation of the moon, it is not the eclipsed but the 
eclipsing body, upon whose circumference we wish to reckon these 
angles, to know at what points the star will undergo immersion 
and emersion, especially the latter. And where two spherical 
bodies are tangent to each other, the distance of the point of 
contact reckoned from the vertex or north point on the limb of 
either, must be 180° different from the same reckoned in the 
same direction on the circumference of the other. For an oc- 
cultation therefore 

N=180°++. 
V = 180 o + + + d. 

To apply these formulas, we have by simple proportion be- 
tween the values of u, v, p—u, and q—v, at 8 h 12 m , 8 h 13 m , and 
10 h 42 m , 10 h 43 m , 





8h 12m 1()\8. 


10h 42m 5i s .3. 




8h 12m ios.8. 


10h 42"> 5R3. 


u 

V 

p—u 
q-v 


4- 1901".l 
-{-2069 .3 
— 1532 .5 
-f-1002 .1 


4- 2425".8 
4-2116 .1 
4-1518 .1 
— 1034 .7 


I. u 

I. V 


4- 3.27900 
4-3.31532 


+ 3.33485 
-f- 3.32554 


I. tan 6 

e 


4-9.96318 
4-42° 34'.5 


4- 10.05931 
4-48° 54'.0 



OOCULTATIONS AND ECLIPSES OF THE SUN. 



113 





8h 12™ 10*.8. 


10h 42"» 51S.3. 


I. {p—u) 
I. (q-v) 


— 3.18540 
+ 3.00091 


+ 3.18130 
— 3.01481 




+ 10.18449 

56° 49'.2 
99o 23'.7 


+ 10.16649 

235° 43'.4 
284° 37'.4 



108. To project an occultation or solar eclipse. 

These results may be exhibited to the eye by a projection, 
analogous to that employed in the case of a lunar eclipse. In 
fig. 26, let AOB and COD, be the circle of declination passing 
through the sun, and a perpendicular to it ; and let p and p—u 
be reckoned on CD, to the left hand when negative, to the right 
when positive ;* also let q and q—v be reckoned on AB, upwards 
or downwards, as the quantities are positive or negative. With 
the radius D (1 — sin *r cos z) from the centre O, describe the 
dotted circle NCC'Z to represent the sun ; and with the radius 
d+D (1 —sin it cos z), describe ACBD. The moon will of course 
just touch the sun, when her apparent place is any where on the 
circle ABCD. 

Set off OU, OU', OU", equal to the three values of p — u re- 
spectively, and perpendicular to these, UV, U'V', U"V", equal 
to the three values of q—v. Then V, V, V", will represent the 
apparent place of the moon with regard to the sun at the three 
times chosen. Through V, V, V", which are not in the same 
straight line, draw the curve MVV'V'M" for the moon's apparent 
path, either by the rule for describing an arc of a circle through 
three points, or graphically with a rapid, but steady hand. The 
first contact will take place at the moment the moon arrives at 
M, and the last, when it reaches M". Draw the perpendicular 
OM' upon the moon's apparent path ; the moon will then be at 
M' at the moment of greatest obscuration, and the times of 
first contact, greatest obscuration, and last contact, may be ap- 
proximately obtained, as in the case of the moon, by comparing 
the spaces MV, V'M', V"M", with the horary spaces VV, V V", — 
noticing, however, that these latter are not quite equal to each 
other. And by describing with radius d, and centres M, M', M", 

* The motion of the moon will then be from right hand to left on the paper, or as 
we see it in the heavens, when moving from west to east. 

15 



114 OCCULTATIONS AND ECLIPSES OF THE SUN. 

three circles representing the moon, the three phases will be 
fully represented.* 

For the farther illustration of our method, suppose we have 
obtained the values of p, q, u, and v, at the beginning, middle, 
and end of an eclipse. Set off the three values of p, namely, 
OP, OP', OP"; and also those of q, namely, Pm, PW, P"m". 
Then m, m', m", will be the points in which the moon would be 
seen in her true relative orbit at these three moments respec- 
tively. Now from P, P', P", set off the values of —u, namely, 
PQ, P'Q', P"Q" ; and let M, M', M" differ from m, m', m", as re- 
ferred to the line AB, by the several values of — v, and join wM, 
m'M', m"M". 

Now at the beginning, middle, and end of the eclipse, the moon 
as seen from the centre of the earth is in its true orbit at m, m', 
and m". A vertical circle as it relates to the hour circle AB, 
lies at those three instants, in the directions m'M., m'M', and m"M", 
and the moon is apparently depressed by parallax on these ver- 
tical circles by the spaces mM, m'M', m"M" respectively. It 
will be seen then, that the general principle of our method con- 
sists simply in referring the moon's place, her parallax, &c, to 
the co-ordinate axes AB and CD passing through the centre of 
the sun. 

On the Practical Uses of Calculating Occultations and Eclipses. 

109. The principal use of determining the times and phases 
of an expected eclipse or occultation, is for the convenience and 
guidance of the observer. For this purpose it is obvious that a 
very accurate calculation is needless ; the times of first and last 
contact to the nearest minute, and the corresponding angles from 
the north point or vertex to the nearest degree, are amply suffi- 
cient. And furthermore, however accurately an eclipse or oc- 
cultation may be calculated, the computed times are liable to 
differ from those observed by several seconds, and sometimes by 
almost l m of time. The minute perturbations of the moon, 
which analysis has not yet reached, render the calculated places 

* If the diagram be held so as to render mM, m'M', ra"M" perpendicular to the 
horizon in succession, the great circles of the sphere, and the relative position of the 
sun and moon, will be as respects the horizon just as they might appear to the spec- 
tator at the times of the three principal phases. 



OCCULTATIONS AND ECLIPSES OF THE SUN. 115 

of the moon in the Naut. Aim. liable to errors whose maximum 
is between I s and 2 s in R.A., and considerably less in Decl. ; and 
it is on this account that all predictions of the times of occulta- 
tions and eclipses are in a slight degree uncertain. An exact 
calculation is therefore out of place and needless, except in a 
preliminary work, for the satisfaction of the learner, or unless it 
is desired to procure results as soon after observation as possible. 
But in the regular economy of astronomical calculation and ob- 
servation, it is found expedient, as regards the saving of time and 
labor, to predict only approximately, and allow observation it- 
self to furnish the exact data for subsequent operations. 

110. When the observer is well prepared with the approxi- 
mate times and angles of immersion and emersion in any eclipse 
or occultation, he may apply himself to the observation of the 
phenomenon, without danger of having his attention weakened 
by being kept too long excited, or divided over a large arc of the 
moon's limb through ignorance of the place. If for greater 
security, he takes the second from the clock, and begins to ob- 
serve two or three minutes before the time, he will not lose his 
count in so short a period, and the ordinary quickness of the eye, 
confined to a small portion of the moon's edge, will be sharpened 
by instant expectation, as the moment approaches. We will 
therefore, (as proposed in Art. 31,) direct the observer how to 
prepare lists of occultations and eclipses for his particular locali- 
ty, in a more expeditious manner than that pointed out in Art. 
104 ; and as we have already calculated an eclipse, we will for 
this purpose choose an occultation. In our chapter " on the 
Transit Instrument," Art. 76, we have introduced among the 
transit observations of Oct. 17, 1839, the immersion and emer- 
sion of 6 Capricorni as observed on the same evening. It is ob- 
vious, that a previous approximate calculation was necessary, 
and to this instance we will therefore direct the student. 

111. Example 2. It is required to find approximately the New 
Haven times of immersion and emersion, and the corresponding 
angles from the moon's vertex, in the occultation of 5 Capricorni, 
Oct. 17, 1839. 

In the Naut. Aim. for 1839, Art. " Occultations," p. 536, we 



116 



OCCULTATIONS AND ECLIPSES OF THE SUN. 



find for this occultation 13 h 19 m = Gr. m. t. of true conj. ; 

21 h 38 ra 12 8 = A ; and —16° 51' = 4. From the Gr. m. t. of true 
conj., 13 h 19 m , we find by eq. Y, (using logarithms to two places 
only,) T = 13 h 35 m . As a third approximation would of course 
tend to increase this value a little, make T = 13 h 40 m ; then 
T— 30 m = 13 h 10 m , and T-f 30 m = 14 h 10 m . 

For the sake of brevity, our process in Art. 104 may be ab- 
breviated in the following ways. Perform the computations for 
the two times 13 h 10 m and 14 h 10 m only, and employ but three 
places of logarithms ; take out J, 8, *, d, to the nearest tenth of 
a minute of space, — a, A, to the nearest second of time, and 
f/,— A to the nearest minute of a degree ; and in doing so, em- 
ploy simple proportion only, interpolation being unnecessary. 

The computation of the term — cos 8 sin J (a— A) 2 may be dis- 

pensed with, and that of the term ie sin <p cos 8 need be perform- 
ed but once, being the same for both times. With these simpli- 
fications, we readily obtain 





P- 


u. 


p—u. 


2- 


v. 


q—v. 


d. 


Oet. 17d 13h 10™ 

14h 10m 


— 4'.6 

+26'.4 


+ 4'.4 
+15'.5 


— 9'.0 
+1C.9 


+28'.2 
-J-41-'.8 


4~49'.0 
^48'.2 


- 20'.8 

- 6'.4 


15'.8 



An easy projection will now afford us the approximations of 
which we are in search. With the centre O, (fig. 27,) and ra- 
dius 15.8, describe the circle ACBD, and draw the two co-ordi- 
nate axes AB, CD. Let the -f-and — values of p—u, q—v, &c, 
be set off in the same directions as in fig. 26. # Set off OU, OIP, 
equal to —9.0, +10.9; and perpendicular to these US, IPS' 
equal to —20.8, —6.4, respectively. Through S and S' draw 
SE for the apparent path of the star ; of course, I will be the 
point of immersion, and E that of emersion, on the moon's limb. 



* In this projection, for the sake of convenience, the moon is supposed to be sta- 
tionary, and the star to move behind it. To correspond with this supposition, the 
directions in which -\- and — values are laid off should be in every way the reverse 
of those in fig. 26. But since the projection is adapted to the situation of the bodies 
as seen in an inverting telescope, the directions of the -\- and — values, remain the 
same as in fig. 26. 

If the above projection be set off on a scale of 10' to the inch, and still more, if 
of 5' to the inch, the results may be obtained with little care, and with sufficient ac- 
curacy 



OCCULT ATIONS AND ECLIPSES OF THE SUN. 117 

81 8 8 

Then 13 h 10 m + ^y = 13 h 10 m +l h x — ^ = 13 h 31 ra .5 = Gr. mean 

time of immersion ; and 14 h 10 m + ^y = 14 h 10 m + l h x — -- = 

14 h 23 m .9 = Gr. mean time of emersion. 

Again, find u and v for the times of immersion and emersion, 
as follows : 





u. 


v. 


Oct. 17d 13* 31™ 5 
14i> 23™ .9 


+ 8'.4 
-t-18'.O 


+48'.7 
-t48'.0 



Set oif OP, OP' equal to +8.4, +18.0 ; and PZ, PZ' equal to 
.+48.7, +48.0 respectively. Then OZ, OZ', will represent the 
directions of a vertical circle, at the times of immersion and 
emersion ; and the vertex of the moon at those instants will be 
the points V and V respectively.* With a protractor measure 
from V and V around the moon's limb to the right hand the arcs 
VCI, and V'CE. These will be found equal to 183° and 279°, 
and are the " angles from moon's vertex" of the points of disap- 
pearance and reappearance. Collecting these results, we have 

13 h 31 m .5— 4 h 51 m .8 = 8 h 39 m .7 . . . N. H. m. time of Immersion. 
14 h 23 m .9-4 h 51 m .8 =9 h 32 m .l . . . N. H. " " Emersion. 

183° = Angle of point of disappearance from vertex. 

279° = " " reappearance " " 

Example 3. Required the approximate New Haven times of 
immersion and emersion, and the corresponding angles from the 
moon's vertex, in the occultation of a Scorpii, (1 mag.) June 20, 
1842. 

Ang. from vertex. 

Time of Immersion . . . . 7 h 33 m .l 355° 

Time of Emersion . . . . 8 h 23 m .4 280° 

* By holding the diagram so that OV and OV' shall be successively perpendicular 
to the horizon, the position of the star as regards the moon at the time of immersion 
and emersion is represented. The observer at the telescope therefore, without hav- 
ing measured the angles instrumentally, may from an inspection of the diagram esti- 
mate the places where the star will immerge and emerge. 



118 METHODS OP DETERMINING 



CHAPTER VIII. 

ON THE METHODS OF DETERMINING THE LATITUDE AND LONGITUDE 

OF PLACES. 

112. The great services which Astronomy has rendered to 
navigation, consist entirely in enabling the mariner to find any 
where on the ocean his latitude and longitude from celestial ob- 
servations. And no less on the main land, the settlement of 
latitudes and longitudes by astronomical means, in sufficient num- 
bers to determine the windings and conformations of coasts, the 
situation of important places, and the boundary lines of different 
countries and states, constitute the only immediate application of 
celestial science to the common wants and demands of civilized 
life. So one of the first objects of the private observer, the 
problem to which he naturally turns his earliest attention, as the 
basis of subsequent researches, is the determination of his own 
latitude and longitude. 

Of the two, the latitude is much the most easily determined, 
since it is reckoned from a fixed circle, the equator, — or in effect, 
from the pole of the heavens, a point fixed in the sphere, and 
whose position relative to the observer's zenith is readily determin- 
ed by simple instruments. The longitude, on the other hand, since 
no one terrestrial meridian is in any way distinguished from all 
others, must be reckoned from one arbitrarily chosen, whose posi- 
tion relative to the observer's meridian is indeterminable by any 
direct method. 

113. Methods of finding the Latitude. 

The latitude of a place is equal to the altitude of the pole, or 
to the complement of the altitude of the equator at that place, 
and may be found in many different ways. We shall endeavor 
to put the student or observer in possession of the principal and 
most useful of these methods, and of the means of reducing the 
observations. It is to be presumed that at this stage of the work, 
he is well versed in the practice of working out results from 
algebraic formulae ; and we shall therefore frequently omit ex- 



THE LATITUDE AND LONGITUDE OF PLACES. 119 

amples, and thus gain room for a more complete exhibition of 
the subject. 

First method. By altitudes of the pole star at any hour of the 
night. — With a sextant and artificial horizon, or other suitable 
instrument, take double altitudes of the pole star, and let them 
be included within a space of 10™ or 20 m in duration, or be 
divided into sets occupying as short intervals of time. Take the 
mean of these double altitudes, and dividing by 2, correct for re- 
fraction by the usual tables ; also let a mean be taken of the 
times corresponding to the several altitudes. The Naut. Aim. 
for each year contains near the end " Tables for determining the 
Latitude by Observations of the Pole Star out of the Meridian," 
and since in the " Explanation of Articles" an example is worked 
out in full, it will be unnecessary to introduce one here. (See 
pp. 563-5, and pp. 602-3, Naut. Aim. for 1839.) 

With an instrument competent to give a distinctly bright 
image of the pole star, this method is probably the best of 
those in common use, since it is practicable at any clear hour 
of the night, and to any degree of repetition of measures ; it is 
also one of the easiest of reduction. 

Second method. By meridian altitudes of the pole star. — The 
true time of the pole star's upper or lower culmination, as indi- 
cated by a good sidereal clock or chronometer, being known, an 
altitude taken at that time will give the latitude at once, after 
the usual corrections, by adding or subtracting the polar distance 
of the star, according as it is at its lower or upper culmination. 
A single altitude, however, is seldom satisfactory; and the pole 
star changes its altitude so slowly, that the mean of observations 
through a space of 15 m on either side of the meridian will not 
give a latitude in error as much as 5" from this cause, — a degree 
of accuracy usually sufficient for the sextant. If the time of 
observation be confined to 5 m on each side of the meridian, the 
latitude will not be affected to the amount of I". 

114. Third method. By meridian altitudes of the sun or other 
heavenly body. — The sun ascends and descends so rapidly, even 
near the meridian, that the mean of two or three altitudes would be 
materially too small. A single altitude may be taken, even with- 
out a knowledge of the time, if the observer, as noon approaches, 



120 METHODS OF DETERMINING 

keeps, for instance, the lower limbs of the two images in exact 
contact as they gradually tend to recede from each other, until 
they recede no longer. The half of the reading, corrected for 
semidiameter, refraction, and parallax, is the true meridian alti- 
tude, whence the latitude is readily found. 

We have, however, a ready means of reduction for altitudes 
observed through a considerable interval before and after noon. 
To the local mean times of observation, reduced from sid. times, 
or taken with a chronometer, apply the " equation of time," (N. 
A., p. II of the month,) corresponding to the Gr. m. times of ob- 
servation, and the results will be the local app. sol. times, or hour 
angles of the sun from the meridian, at which the altitudes were 
taken. Now the versed sine of the sun's arc or hour angle, for 
a short period, varies very nearly as the number of seconds of 
his descent. Take the mean of the nat. versines* of the several 
hour angles from noon ; this mean will bear the same ratio to 
the mean of the corrections of the sun's altitude, as any one of 
the versines to the corresponding correction. This ratio is ex- 
pressed by the factor of reduction 

cos 8 cos cp cosec z cosec l",f 
by which the mean of the versines being multiplied, we obtain 
the mean or final correction of the altitudes. The symbols <5, <p, 
and z, signify as usual, the deck, latitude, and zen. dist. of the 
body. 

Ex. 1. It is required to deduce the latitude of New Haven 
from the following altitudes of the sun, and corresponding mean 
times of observation, on the noon of June 30, 1840; — the chro- 
nometer being fast of N. H. mean time 4 h 50 m 50 s .7. 

In the following calculation, it is sufficient to take out 8, 9, and 
z to the nearest minute of space, and make z = q>—8. The sun's 
Deck, 23° 9' 53", is to be taken from the Naut. Aim. for the Gr. 
app. times of observation. By observing the upper and lower 
limbs alternately, any correction for semidiameter becomes un- 

* If the logarithmic tables do not contain nat. versines, subtract nat. cos from 
unity. 

t For want of room, we have omitted any demonstration, which the student can, 
however, supply without much difficulty, with the aid of a figure of the celestial 
sphere. , 



THE LATITUDE AND LONGITUDE OF PLACES. 



121 



necessary ; if, however, it should be required, it may be found in 
the N. A., p. II of the month. 



Double 


Chron. 


N. H. 


Equation 


N. H. 

Apparent 

Times. 


Sun's hour 


Versines 


the sun. 


Times. 


Times. 


of Time. 


angle in arc. 


of arcs. 


143° 37' 13" 


4h 47m 50s 


— 3« 1* 


-3«> 19» 


-6 m 20s 


-1° 35' 0" 


382 


" 42 15 


" 51 42 


-f-0 51 


It 14 


2 28 


37 


53 


" 43 21 


" 53 35 


2 44 


" " 


-0 35 


-0 8 45 


3 


" 42 47 


" 56 6 


5 15 


(( I. 


+1 56 


+0 29 


36 


" 40 34 


" 58 19 


7 28 


<( M 


4 9 


1 2 15 


164 


» 32 16 


5 2 52 


+12 1 


tt (( 


+8 42 


+2 10 30 


720 



238' 26" 



6) 1363 



2) 143° 39' 44" 

71 o 49' 52" 

-20" 

+ 3" 

-f-1' 44" 

71° 51' 19" 



. App. altitude.* 
Corr. for refr. 
Corr. for par. 
Corr. for mean of alts, by 

the above process. 
True altitude. 



J = +28° 10'... cos. 
= 41° 18' . . . cos. 
z= 18° 8' . . . cosec.t 
cosec 1" 



227 

9.96348 
9.87579 
10.50692 
5.31443 



5.66062 
227 (+4 to Index)! 6.35603 

104" = corr. for 2.01665 

mean of meridian altitudes. 



18° 8' 41" . . . True zen. dist. 
23° 9' 53" . . . Sun's decl. 



41° 18' 34" . . . Latitude of New Haven. 

Fourth method. By double altitudes of the sun or of two stars. — 
By taking two altitudes of the sun at times differing from each 
other an hour or more, both the latitude and the time may be 
determined, without previous knowledge of either. The space 
of time between noon and the observation nearest noon, should 
generally be less than that between the observations. 

Let P (fig. 28,) be the pole, Z the zenith, S the place of the 
sun at the first observation, and S' at the second. Find the app. 
sol. times of the observations as in the last example, and subtract 
the first from the last ; then since the motion of the sun in R.A. 
is the measure of solar time, the difference of these two times 



* These altitudes are too great for the sextant, but may be taken with a reflecting 
circle, used as a sextant. 

t If the cosec. is not in the tables, take the arith. comp. of the sine. 

t The numbers in the column of versines, taken from the tables to 6 places of de- 
cimals, are of course millionths of radius. Since radius is supposed to be divided 
into 10,000,000,000 parts, if the columnar numbers are considered as integers, 4 must 
be added to the index of the logarithm. 

16 



122 METHODS OF DETERMINING 

equals the hour angle SPS'. The sun's Decls. being taken out 
for each of the times, PS and PS', their complements, are also 
known. ZS and ZS' likewise, are the complements of the cor- 
rected altitudes. Therefore, in the triangle SPS', having L SPS' 
and the adjacent sides given, we may find first L PS'S, and then 
the side SS'. Again, in the triangle SZS', with the three sides, 
we obtain L ZS'S, and subtracting from this ZPS'S, we have 
L PS'Z. Lastly, in the triangle PZS', with the L PS'Z, and the 
adjacent sides, we may find PZ, which is the complement of the 
latitude. 

Let 5 and 8' be the Decls. of the sun at S and S', z and z' the 
zen. distances ZS and ZS' ; also let the hour angle SPS'=A, 
L PS'S = «, L ZS'S = ft SS' = tf, and the latitude = 9. Then 
the formulae necessary for the work will be as follows : 

1. tan x* =cotan 6 cos h.-\ 

2. tan a — cosec \ 90° - (oc-\-S') \ tan h sin x. 

3. sin d = cosec a sin h cos <5. 

4. y = i (z+z'+tf). 

5. cos \ (3 = Vsin y sin (y - z) cosec z' cosec tf. 

6. tan v — cotan 5' cos (/3 - a). 

7. sin <pj == sec v cos (z' -^ v) sin 6'. 

Single altitudes of two bodies are equivalent to double alti- 
tudes of a single body, separated by a considerable interval of 
time ; the sun and moon may be employed in the day time, and 
any two bright stars near the equator, and differing l h or more in 

* x is like or unlike 90°— <5, as h is > or < than 90° ; and a is like or unlike h, 
as x is > or < than 90°— S. Again, v is like or unlike 90°— 8, and <p is like or un- 
like z -+■ v, according as (3 -«- a is > or < than 90°. 

t Remarks on the Formula. — These are all from Hutton's Tables, London edition, 
1834. For Nos. 1 and 2, see page xlii of Introduction, case 3, first formulae ; No. 3, 
page xliii, case 6, first formula; No. 5, page xlii, case 1, last formula; Nos. G and 7, 
page xlii, case 3, nearly the last formulas. The only changes in our formulae are 
easily recognized ; for instance, in the 7th, (making a' =: pol. dist. PS, or complement 
of (3',) — the proportion of Hutton, page xlii, case 3, . . . cos v : cos (z' ^ v) : : cos A' : cos 

COS (% *^* 75^ COS A' 

(90° — 0) (or PZ) gives by multiplication, cos (90° — ^)= . But 

cos (90°— 0) and cos A' are equal to sin <p and sin 6', and cos v may be struck from 
the denominator, by inserting its reciprocal, sec v, in the numerator. It thus be- 
comes identical with our formula 7th. 

t For a shorter process of finding 0, — substitute the rule I, p. lx, Hutton, for for- 
mulae 6th and 7th. 



THE LATITUDE AND LONGITUDE OF PLACES. 123 

R.A., are suitable objects for night observation. The sid. time 
elapsed between the two observations must be added to or sub- 
tracted from the dim of the R.A.'s of the two stars, according 
as the star of greatest or least R.A. was the first in order of ob- 
servation. The sum or difference will be the hour angle h, and 
the process of reduction will be the same as before by the above 
formulae. 

As we have in this chapter considered altitudes as taken by 
unfixed instruments, such as the sextant, and its different modifi- 
cations of reflecting and repeating circles, we shall depart so far 
from the subject of the chapter, as to give the observer a formula, 
by which he may determine his time from a single altitude, the 
latitude being known. In the triangle PZS, (fig. 28,) the three 
sides are known, and to find the hour angle ZPS = h, we have 
the formulae, 

sin i h = sf sin \ (%+(<?— <5) sin £ (z — ((p — 6) sec <p sec 6* 
R.A. of sun or star + h\ — sid. time of observation. 

115. Methods of finding the Longitude. 

The principles of several ways of determining the longitude 
are illustrated in Olmsted's Astronomy, Arts. 272-278, so as to 
need little general explanation here. "VVe will classify the nu- 
merous methods as follows : 

1st. Such as show the difference of local times between any 
two places, by an event occurring at the same instant of absolute 
time to both. 

Of this class are lunar eclipses, which, however, are uncer- 
tain to many seconds, by reason of great indefiniteness in the 
earth's shadow. Also the eclipses of Jupiter's satellites; but 
these also give a longitude uncertain to many seconds, because 
the satellites are bodies of considerable size, and disappear and 
reappear gradually. Yet from the frequency of these eclipses, 
observations may be multiplied to such an extent, as to give a 
tolerably accurate result. The Naut. Almanac, at p. XX of each 

* Baily's Ast. Tables and Formula?, Form. XV, p. 89. Hutton's Tables, p. xlii, 
case. I, first formula. 

t h is always reckoned by hours, in the direction of the apparent revolution of the 
heavens, and ranges from h to XII h on the west side of the meridian, and from Xll h 
to XXIV h or 0»» on the east. 



124 METHODS OF DETERMINING 

month, furnishes a list of Greenwich immersions and emersions, 
to which ready reference may be made by the observer. 

Belonging to this class are coincident observations, at different 
places, on meteors, the explosion of rockets, the alternate appear- 
ance and disappearance of strong intermitted lights, &c. These 
are almost instantaneous occurrences, and hence longitudes may 
be obtained from them to a higher degree of accuracy than by 
any other method. The application of these contrivances is, 
however, restricted, of course, to small differences of place. 

The method by chronometers indirectly belongs to this class, 
since the arrival of the pointers of a Greenwich chronometer to 
h m s at New Haven marks the occurrence of Greenwich 
mean noon at that instant, and the local time of New Haven can 
be noted at the same instant of absolute time. This method is 
susceptible of much greater accuracy than those of eclipses of 
the moon and of Jupiter's satellites, especially for short dis- 
tances of transportation.* (See further, Olmsted's Astronomy, 
Art. 174.) 

116. 2nd. Such as show the difference of local times between 
the places, by corresponding observations on a phenomenon not 
happening at exactly the same instant of absolute time to both, 
but at instants differing by an amount calculable from known 
data. 

Occultations of stars and planets, and eclipses of the sun, are of 
this description. The immersion or emersion of a star may oc- 
cur at instants of absolute time differing by two or more hours 
at two places, because the parallax of the moon displaces her dif- 
ferently for different observers ; but, by the exact calculation and 
application of those displacements, an observation at any one 
place may be reduced to such as it would be at the centre of the 
earth, and it may then be compared with an observation at any 
other place, which has been reduced in a similar manner. 

* The recent introduction of Atlantic Steam-ships, and their rapid passages from 
land to land, has given rise to several novel experiments on determining differences 
of longitude between stations on the two continents, by the transportation of chro- 
nometers. The first three or four attempts have been very successful, and the lon- 
gitude of New York, and incidentally that of New Haven, from Greenwich, has 
been well settled. 



THE LATITUDE AND LONGITUDE OF PLACES. 125 

The problem of finding the longitude from an observed stellar 
or solar occultation, is nearly the reverse of that for finding the 
times of an occultation, at a place of known longitude. Sup- 
pose, for example, we take the observed time of immersion 
of 8 Capricorni, as reduced to true sidereal time on page 69, 
Art. 76. The corresponding instant of mean time, when reduced 
to Greenwich mean time, by applying the assumed difference of 
longitude of New Haven, (4 h 51 m 46 s ), in case the assumed lon- 
gitude is incorrect, will not be the true Greenwich mean time of 
immersion, but will be in error to the same amount as the as- 
sumed longitude. Now if we calculate equation (A), Art. 103, 
for this erroneous Gr. time, a time not exactly agreeing with that 
of the observed immersion, it is evident from the conditions of 
the equation that the two members cannot be quite equal to one 
another ; and from the calculated difference between them, we in- 
tend to deduce the unknown error of the assumed longitude. Let 
us examine what parts of (A.) will be affected by the error. In 
the first place, the true local sid. time of immersion (fx) is furnish- 
ed at once by observation, and requires scarcely an approximate 
knowledge of the longitude. Now the terms u and v, as will be 
seen by their expanded values, depend on the quantity jx, and on 
others either unchanging, or changing imperceptibly in a very 
short interval of time ; and therefore a small error in the as- 
sumed longitude does not affect them. But a and 8 are of rapid 
change, and, if taken out for a slightly erroneous Gr. time, ren- 
der the differences a.— A, 8—4, and consequently the terms p and 
q incorrect. If, therefore, we find for the assumed Gr. m. time 
of immersion, . . . (p — u) 2 + (q — v) 2 —d 2 = 2 E, the above remark 
renders it evident that the terms u and v are the correct values 
belonging to the instant of immersion, while p and q are values 
corresponding to a time as many seconds before or after the in- 
stant of immersion, as the assumed longitude is in error. 

Let t be the true mean time of the observation at New Haven, 
I the assumed longitude, and I + x the true unknown longitude ; 
x to be expressed in parts of an hour. Also let p' and q' be the 
hourly variations of p and q. After calculating 2 E from the 
equation in the last paragraph, we may find x by the formula, 

~ (p—u) P' + {q-v) 9'' 



126 METHODS OF DETERMINING 

This equation would show the exact correction of the assumed 
longitude, if we could rely on the perfect accuracy of the Naut. 
Almanac. But as we have already remarked in Art. 109, the cal- 
culated place of the moon is liable to differ a little from the ob- 
served place. The errors of a and 8, affect those of p and q, 
and thus vitiate the result. If e is the minute fraction of an 
hour, by which the immersion, for instance, is delayed beyond 
the calculated time, on account of the errors of the tabular 
values of a and d, then 

x + e = E 

(p-u) p' + (q— v) q'' 

Now, if the occultation is observed any where else, a similar 
equation may be formed for the second station, and since e is 
constant at least for a few hours, we can eliminate it by subtrac- 
tion, and obtain the true difference of longitude between the two 
places. 

117. Calculation of the longitude from an observed occultation. 

Ex. 2. Assuming the longitude of Philadelphia to be 5 h m 42 s , 
and that of New Haven 4 h 51 m 46 s , it is required to deduce the 
difference of longitude between these places from the following 

* For t-\-l-\-x is the true moment of immersion; and while u and v are values 
corresponding to this time, p and q are values corresponding to the time t-{-l. Let 
p, and q, be the unknown values answering to the time t -f- 1 -f- x ; then p,—p and q, — q 
are the variations of p and q for the fraction of an hour x ; and since these variations 
are proportioned to the time of change, 

c 1 -; p, - q , - lh -*- 

Again, since p n q„ it, v, are values belonging to t-\-l-\-x, the true moment of im- 
mersion, 

(p l -uY+(q l -v)^-d^ = 0. 
(2>— m) 8 -|-(7 -i;) 8 -d2 = 2E. 
Subtracting the lower from the upper, 

■(Pr-P) (fl+J»-2u)-{-( ?/ - ? ) (£ + 7-99) = -&E. 
Or since in this differential equation, p,-\-p, when x is small, is very nearly equal 
to 2p, and qi-\-q to 2q, 

(p-u) (p,-p)-\-(q-v) ( ?/ -5) = -E. 
Dividing by eq. (1.), 

E 



and (2.) x = - 



x 
E 



(P-U)p' +(?-»)?'' 



THE LATITUDE AND LONGITUDE OF PLACES. 



127 



corresponding observations on the immersion of 8 Capricorm, Oct. 
17, 1839. 

Imm. at Phil 22 h ll m 37 s .68 . . . Phil 

sid. time. 
N.H.. (See Art. 76, pp. 65-9,) . . 22 23 37 .21 . . N. H. 

sid. time. 

It is evident at once that M- = 22 h ll m 37 s .68 for Philadelphia, 

and 22 h 23 m 37 s .21 for New Haven. Adding to each of these 

the respective longitude of the two places, as assumed above, 

and converting from Gr. sid. to Gr. m. time, we have 

A An .. V. • ( 13 h 28 m 53 s .50... for Phil. 

Assumed Gr. m. time of immersion, \ m , „ 

(13 31 56 .53... for N.H. 

Calculating for these times equations (C), (D.), (E.) and (F.) 

in Art. 103, we have, 





Gr. to. time of calc. 


P- 


u. 


p—u. 


?• 


v. 


q— v. 


For Phil. 
» N. H. 


13h 28" 53.50 
13 31 56.53 


+31 I'M 

+405 .6 


+387".3 
+514 .2 


- 76".2 
-108 .6 


+1945".2 

+ 1986 .6 


+2S87".4 
+2924 .8 


-942".2 
-938 .2 



Here in the progress of 3 m 3 S .03, p undergoes a variation of 
+94". 5, and q of 41". 4 ; therefore, their hourly variations (which, 
for the determination of a few seconds of change, may be safely 
deduced in this rude way,) are as follows: p'=+1858".7; 
2' = + 814".3. Then, 

For Phil. (p-u) 2 +(q-v) 2 -d* = 2E = -1748" .;. E=- 874". 
For N. II. " =2E = -3250" /. E=-1625". 

Again we have 

For Phil. - 7 x , , ^— = x + e = -3 S .46. 

(p—u)p' + (q-v) q' 

For N.H. " = x -\-e = — 6 S .06. 

Giving the N. H. quantities an accent, to distinguish them from 
those obtained for Philadelphia, 

I + x + e = 5 h m 42 s — 3 8 .46 = 5 h m 38 s .54. 
l'+x'+e=4 51 46-6.06 = 451 39.94. 
Since e is the same in both, we obtain by subtraction, 

(l>±x')-(l + x) = -8 m 58 s .60 . . . True diff. of longi- 
tude of N. H. from Phil. 

If the longitude of Philadelphia is well determined in com- 
parison with that of New Haven, and may be considered for our 



128 METHODS OF DETERMINING 

purpose as needing no correction, then, making x = 0, we have 
e = —3.46, and by substitution 

1>+ X > = 4: h 51 m 43 s .40 . . . True longitude of New Haven. 

118. 3rd. We bring under a third class such methods of find- 
ing the difference of longitude, as depend on the measurement 
of the moon's angular motion in the interval between two in- 
stants of time ; the two observed instants being in the local times 
of the two places respectively. 

The solution of these methods depends on this simple princi- 
ple, — that as we can calculate the hourly motions of the moon 
for any given time in given directions, we can readily deduce 
the absolute time of her describing any known and measured arc. 
But the absolute time elapsing between any two instants, ex- 
pressed in the local times of the two places, is all that is neces- 
sary to determine their difference of longitude. 

The method of lunar distances, and that of moon-culminating 
stars belong to this class. The mode of taking the distance of 
the moon from a star or the sun by the sextant, has been ex- 
plained in the chapter on that instrument, Art. 43. Such an ob- 
servation gives the distance from the apparent edge of the moon 
to the app. place of the star, — whereas the distance between 
their true places is wanted. Preliminary to the formulae for ob- 
taining the true distance, the app. alts, or zen. distances of the 
bodies must be reduced from the true ones, and the observed 
angle must be corrected for the moon's augmented semidiameter. 
For the true altitude — of the moon, for instance, take out for 
that body a and 8 from the Naut. Aim. ; then ^ (the sid. time of 
the observation) — a = h, the hour angle of the moon from the 
meridian. In the triangle PZS, (fig. 28,) we have ZPS = h, and 
the adjacent sides are the complement of <p and S ; the formulas, 
of course, are like Nos. 6 and 7, in the " Fourth method of find- 
ing the latitude." 

tan x = cotan <p cos h. 

cos z = sec x cos \ 90° ^ (x-\-S) \ sin 9. 

The true is now to be converted into the app. zen. dist. by 
applying the parallax and refraction. Call the moon's par. in 
alt. n, and finding the reduced horizontal parallax *, as in the 
chapter " on Occultations ," &c., make 



THE LATITUDE AND LONGITUDE OF PLACES. 129 

1st approx. of parallax = if sin z. 
Then n =if sin (z + 1st approx. of parallax). 
And so for the sun or a star, except that only one approxima- 
tion of parallax is needed for the former, and none for the latter. 
The next step is to apply the moon's augmented semidiameter 
to the observed angle. Calling the true semid. d, the aug- 
mented semid. = -A — d sin (z+U) cosec z. If the sun 

sin z 

is one of the bodies observed, the semidiameter of that body 
also is to be added to the observed angle to obtain the app. dist. 
of centres. Call the app. and true altitudes of the moon A and 
A' ; the same quantities for the sun or star a and a! ; and the 
app. and true distances between the bodies A and A'. We are 
now prepared to find A ' by the following formulae : 

x = \ (A+A+a). 



(cos x cos (a?— A) cos A' cos a' sec A sec a)' 

sin y = * 2 L 

J cos h (A' -fa') 



1 



sin i A'=cos i (A'+a') cos y. 

We have now the angular distance as it would appear at the 
centre of the earth at the local time of observation. The Naut. 
Aim. gives this distance for intervals of three hours of Gr. time, 
and we may find at what moment of Gr. time the true central 
distance was the same as the calculated, by interpolating between 
these values. And having both the Gr. and local times at which 
the moon was at a certain true central distance from a star, the 
longitude is at once determinable. 

The method of moon-culminating stars consists in measuring 
by the transit instrument the arc of R.A. described by the moon 
in passing from the Greenwich to another meridian. Let 7, as 
before, be the assumed longitude of New Haven, or any other 
place, and l+x the true longitude to be determined. Let a and 
a! be the true R.A.'s of the moon at the moments of its passing 
the Gr. and N. H. meridians respectively. These will be the 
sidereal times of her transit at those places, because on the me- 
ridian the depression of parallax does not affect her right ascen- 

* The demonstration of these formulce being omitted for want of space, the reader 
may be referred for authority to " Baily's Tables and Formulae," Form. XLVIII. 
p. 121. 

17 



130 METHODS OF DETERMINING 

sion. In the Naut. Aim., under the head of " Moon-Culminating 
Stars," are given for every day in the year, the moon's R.A., and 
" var. for 1 hour of longitude," both at her upper or visible, and 
lower or invisible passage, across the meridian of Greenwich. 
These quantities being thus calculated for every 12 hours of Ion- 
gitude, we can interpolate their values for any intermediate lon- 
gitude, in the same way as we interpolate between equal intervals 
of time. Find therefore by interpolation the moon's R.A., and 
var. of R.A. in 1 hour of long., for the assumed longitude I, and 
call them a" and a'". Now a', the sid. time of the moon's tran- 
sit at N. H., is of course her R.A. for the true longitude l+x ; 
and consequently a! — a!' is the increase of the moon's R.A. for 
the small space of longitude x, immediately after passing the me- 
ridian I. Then a.'" (the increase for 1 hour of longitude at the 
meridian I) : a' — a" : : l h of longitude : x ; and we have x in parts 
of an hour of longitude by the following equation : 



118. Calculation of the longitude from an observed moon-cul- 
mination. 

Ex. 3. The assumed longitude of New Haven being 4 h 51 m 46 s , 
it is required to deduce the true longitude from the observed me- 
ridian passage -of the moon at New Haven on the 11 th of Oct. 
1839, as given in the transit observations of that evening, (Art. 
76.) 

We find among the equations for stars in the column eqs. I, 
p. 66, an equation for the moon, — which on reduction (eqs. Ill, 
p. 68,) gives as its result . . . — 9 m 57 s .51, for the apparent correc- 
tion (x) of the clock, — the R.A. of her Gr. passage being as- 
sumed for that at New Haven. The true correction of the clock, 
(+53 s ... 4 s ), as obtained from stars, might be applied to this quan- 
tity, according to the exam, on p. 69, to determine the increase 
of R.A. ; but it is preferable to deduce this correction from the 
four moon-culminating stars of the Naut. Aim. only, which are, 
for reasons assigned in that work, peculiarly favorable for com- 
parison with the moon. Taking therefore the mean of the cor- 
rections (x), and of the corresponding times, for *] Capr., s Capr., 
5 Capr., and « Aquarii, we have the correction # = +53M2 at 



THE LATITUDE AND LONGITUDE OF PLACES. 



131 



21 h 24 m . The moon crosses the N. H. meridian about 12 m af- 
ter this, and therefore by the exam, on p. 69, a correction of 
S .08 must be added to +53M2, making it +53 s .20. The differ- 
ence between this correction by the moon-culminating stars, and 
the erroneous correction caused by the advance of the moon in 
R.A., is of course the moon's true increase of R.A. in her pas- 
sage between the two meridians = a' — a = + 10 m 50 s .71. We 
are now to interpolate from the Naut. Aim. for the quantity a", 
and to do this with sufficient accuracy, we must calculate the for- 
mula on p. 72, as far as the 3rd or 4th term : 



Oct. 1839. 


a. 


d'. 


d". 


d'". 


17h Qm 


21»> 25-" R66 


+ 26"> 4()s.70 






» 12 


" 51 42.36 


" 26 .22 


— 14*.48 


+ 6*.61 


18 


22 18 8.58 


+ ■ 18.35 


— 7.87 




" 12 


" 44 26 .93 









4h 51m 46 S 
< = — 247— =01.4052. 

df = 1600".7 ; df'= - 14*.48 ; d"> = + 6«.61. 



21h25m K66'=o. 
+10 48 .65 = -\-d't. 



+1.75 



■ d"t 



t-1 



+ .43 = + «ti.'- i ?. 



21'> 35«> 52^.49 



+ 10 50 .83 = a"- a. 
+ 10 50 .71 = a' -a. 



■0».12 = o'-a" 



For a'", from the values 134M5, 132 s .71, 13P.76, and their 
differences,— we have 134 s . 15 - 8 .58 - S .08 = 133 s .50 = a!". 
Then 

a' — a " —0.12 



== — d .000899 



3 S .24 



j\ 



a'" 133.50 

I + x = 4 h 51 m 4G 9 — 3 S .24 = 4 h 51 m 42 s .76 . . . Longitude of N. H. 
If by a similar observation at Greenwich, or other foreign 
observatory, the R.A. of the moon in the Naut. Aim. is found to 
be too great by the quantity a„ the equation becomes 



x-\-c = 



a '_ a "+ a/ 



132 METHODS OF DETERMINING 

One peculiarity of the methods belonging to this class is, — that 
an error in the measurement will produce one between 20 and 30 
times as great in the longitude. Thus, an error in observing the 
transit of the moon of only I s at either station, would make a 
difference, on the average, of more than 26 s ; and an observer 
who takes a lunar distance 10" too small, makes a difference in 
his longitude of about 18 s of time. The reason of this is, that the 
moon's monthly motion is measured by means of sidereal time, 
whose rate of revolution is nearly 27 times more swift. But 
where a distinct phenomenon is observed, as in all the methods 
under the first two classes, no such multiplication of error can 
take place. 

119. We have now arrived at the conclusion of our work, — 
and with this partial review of the terrestrial contrivances and 
means by which astronomers have acquired their knowledge of 
the celestial bodies, we shall turn with increased pleasure to the 
consideration of the results of their labors, which constitute the 
departments of descriptive and physical astronomy. The stu- 
dent would regard, for example, with none the less interest the 
return of the long expected comet, and the exact verification of 
the calculations of mathematicians, because he was acquainted 
with the means by which the observer tracked the body in its 
passage through the heavens, and with his diminutive, but re- 
fined apparatus, recorded unerringly the data from which the 
physical astronomer should predict, without hazard of failure, 
the exact positions which it should in future assume. Nor will 
an eclipse be viewed with less pleasure and satisfaction, after he 
has become able to foretell its time and aspect. And all the data 
of astronomy, in their immensity of extent,— -its processes, and 
magnificent conclusions, will now seem to him far more stable 
and secure, for he has derived confidence from penetrating to 
the very basis of the science, from reviewing the delicate re- 
sources of the observer in his instruments, and the processes of 
observation, until he has arrived at valuable measurements and 
primitive data. 

The observer, too, who has the means of deriving results of 
his own from the heavens, needs no stimulus to prosecute a 
study, which few, who thoroughly engage in it, will easily re- 



THE LATITUDE AND LONGITUDE OF PLACES. 133 

linquish. Every failure of agreement in his conclusions will 
but urge him patiently to solve the difficulty ; every instance of 
success will inspire him with fresh ardor and enterprise : and he 
will find no pursuit more constantly beckoning him forward to 
what lies beyond him, more absorbing in its prosecution, more 
elevating to the mind, or impressing him with a deeper sense of 
the power and wisdom of the Creator 



NOTE TO ART. 95. 

Being a simple Geometrical and Algebraic demonstration of the Formula for 
Occultations on page 93. 

Preliminary to this investigation, we shall introduce three formulae necessary to the 
following demonstration, and shall refer for authority to Young's Spherical Trigonometry, 
a common elementary work and text-book. 

In a triangle ABC right angled at C, 
Formula 1. % cos c = cos a cos b. 

(See Young's Spher. Trig., Art. 52, eq. 3 of right angled triangles.) 
In an oblique angled triangle, 
Formula 2. sin a sin B =. sin b sin A. 

(For . . . (Y. S. T., Art. 52, eq. 1,) -: = -: — r ; and clearing of fractions, we 

v ^ sin a < sin b ° ' 

have formula 2.) 
Formula 3. sin c cos A = cos a sin 6 — sin a cos 6 cos C. 

(For . . . (Y. S. T., Art. 48, eq. 1,) cos a sin b = sin a cos b cos C -\- sin c cos A ; 
and transposing, we have formula 3.) 

Let O (fig. 29,) be the centre of the earth, S the place of a spectator on its surface, and 
M the position of the moon in space. Suppose with the centre O and radius OM, a 
spherical surface to be described, in which let us take X, Y, Z, any points 90° distant 
from each other, and join them by the quadrantal arcs XY, YZ, ZX ; join also OX, OY, 
OZ, OM, and produce OS to meet the sphere in A ; A will then represent on the sphere 
the geocentric zenith of the spectator, and M the true place of the moon. Through O 
draw OM' parallel to SM, and complete the parallelogram OSMN ; then since SM is the 
apparent direction of the moon from the spectator, the observer will refer the moon on an 
infinite sphere to a point corresponding to M', in the same direction from the centre of the 
earth that M is from S. M' therefore represents the apparent place of the moon on the 
sphere. We have then three planes, XOY, YOZ, ZOX, at right angles to each other, and 
to which we can refer the points M, S, and N. 

From M, S, and N, draw MD, SB, and NC perpendicular to OX ; then will OC = BD. 
For if we suppose three planes, P, P', P", passing through the points M, S, and N, parallel 
to YOZ, and cutting OX at right angles, — MD, SB, and NC will lie in those planes, be- 
cause they are at right angles to OX ; the planes will therefore cut OX in the points 
D, B, and C respectively, and DB will be the perpendicular distance between the planes 
P and P', and CO between the planes P" and YOZ. But SM unites the planes P and P', 
and ON the planes P" and YOZ, and these lines, being parallel, have the same inclina- 
tion to their respective planes ; if this anglt. of inclination be called i, the perpendicular 
distance between P and P' (DB) = SM sin <, and that between P" and YOZ (OC) = ON 
sin i. But SM and ON are equal ; therefore OC = DB. 

Since OCN, OBS, and ODM are right angles, OC = ON cos CON = ON cos XM' ; 
OB = OS cos XA ; and OD = OM cos XM. But OD = OB-j-BD = OB + OC, and 
therefore 

I. OM cos XM = OS cos XA+ SM cos XM'. 

By referring S, M, and N, to the plane ZOX by means of the line OY, precisely as we 



NOTE. 135 

have already referred them to YOZ by means of the line OX, we shall obtain the simi- 
lar equation 

II. OM cos YM = OS cos YA + SM cos YM'. 
And referring them to the plane XOY by the line OZ, 

III. OM cos ZM = OS cos ZA + SM cos ZM'. 

Now in the right angled triangle YHM, we have (formula 1.), cos YM == cos HM cos 
YH. For the same reason, cos YA = cos I A cos YI; and cos YM' = cos GM' cos YG. 

Again, by the same formula, cos ZM = cos HM cos ZH = cos HM sin YH; so cos 
ZA = cos IA sin YI, and cos ZM' = cos GM' sin YG. 

Substituting these values in the two last equations, — equations (I.), (II.) and (III.), after 
transposition, become as follows : 

(1.) S3I sin GM' = OM sin HM - OS sin IA 

(2.) SM cos GM' cos YG = OM cos HM cos YH - OS cos IA cos YI. 

/3.) SM cos GM' sin YG = OM cos HM sin YH - OS cos IA sin YI. 

With M', the apparent place of the moon for the centre, and with radius M'T = the 
moon's apparent semidiameter, describe a circle representing the moon as it appears in 
situation and magnitude on the infinite sphere to the observer. Let T be a star distant 
from the moon's centre by her apparent semidiameter ; that is, apparently in contact with 
the moon's limb, and therefore undergoing either immersion or emersion. Then in the 
spherical triangle XTM', we have 

(4.) sin M'T cos XTM' = cos XM' sin XT -sin XM' cos XT cos M'XT. (Form. 3.) 

(5.) sin M'T sin XTM' = sin XM' sin M'XT (Form. 2.) 

Let us now make 
SM r= a = distance of the moon from the place of observation in parts of 031 as ra- 
dius or unity. 
OS = sin 7r = distance of the observer from the centre of the earth in parts of OM as 
radius ; tt being the eq. hor. par. of the moon, reduced in the ratio 
OS 





earth's 


eq. 


rad." 






YH = a 












HM = 6 








Let also 




YG = a' 








YL = A 




GM'=5 








LT = a 




YI=„ 








MT = d> 




IA = 








Z.XTM'— r, 


. . when a' <![ A, 








= 360°-r, . . 


. . when a' > A. 










/.MXT=-( fl '-A), . 


. . when a' < A. 










= (a'-A), . 


. . when a' > A. 



By the substitution of these symbols, equations (1.), (2.) and (3.) become 
(1'.) A sin b' = sin 6 — sin n- sin <p. 

(2'.) A cos 6' cos a'= cos <5 cos a— sin n cos <p cos //. 

(3'.) A cos <5' sin a' = cos <5 sin a — sin -n cos sin //. 

And changing the notation of eqs. (4.) and (5.) in the same manner, 

(4'.) sin d' cos r = sin 6' cos A -cos <5' sin A cos (a'— A). 

(5'.) sin d' sin r = -cos 6' sin (a'— A). 

The angle r is the angle included between the meridian XT and the great circle TM 
passing through the star T and moon M', — reckoned from 0° to 300° in the direction XOY, 
or so that r shall be between 0° and 180°, when a' < A, and between 180° and 360°, 
when a' > A. 

The next step will be to transform in eqs. (4'.) and (5'.), the apparent semidiameter, 
R.A., and Decl. of the moon, into the true ones,— by combination with eqs. (1'.), (2'.) and 
(3'.) For eq. (4'.) the process will be as follows : 



136 NOTE. 

Multiplying by A, and expanding cos (a'-A) . . . Day's Trig. Anal., Art. 208, IV.; 
A sin d' cos r = A sin 8' cos A —A cos &' sm A cos «' cos A 

— A cos 8' sin A sin a' sin A. 

Here A sm <5' occurs in the first term of the second member, A cos 8' cos a' in the 
second, and A cos 8' sin a' in the third ; substituting the values of these quantities as 
given in eqs. (1'.), (2'.) and (3'.) : 

A sin d! cos r = sin 8 cos A— sin -it sin <p cos A 

— cos 8 sin A cos a cos A -(- sin 7r cos sin A cos ft cos A 

— cos 8 sin A sin a sin A -|-sin it cos <p sin A sin /* sin A. 
Recombining by Art. 208, IV., Day's Anal. Trig. : 

A sin d' cos r = sin 8 cos A— cos 8 sin A cos (a— A) 

— sin 7r sin <p cos A -\- sin -a cos sin A cos (/x — A). 
And since in Art. 94 we have proved that A sin d'= sin d; 

(7.) sin d cos r = sin 8 cos A— cos 8 sin A cos (a— A) 

— sin n | sin <p cOs A — cos (p sin A cos (/* — A) j. 
For eq. (5'.), a similar, but shorter process brings out this equation : 

(8.) sin d sin r = —cos 8 sin (a— A) -f- cos sin tt sin (p - A). 
Squaring each of these equations, and adding their squares, (since sin 2 d sin 2 r -j- sin 2 d 
oos 2 r = sin 2 d,) ; 

(I.) sin 2 dt= ^ cos 5 sin (a— A) — sin n cos sin (/<— A) | 2 
-J- < sin 5 cos A — cos 6 sin A cos (a— A) 

— sin 7T | sin cos A— cos sin A cos Qn —A) j j 2 . 
which equation is the same with the expanded formula for occupations in Art. 96, p. 93. 

The formula just demonstrated is, however, far more general than the other. The 
points X, Y, Z, were chosen any three points 90° distant from each other, and our equa- 
tion is therefore universal, and may be practically applied to any great circle of the hea- 
vens, and its pole. The equator is the one to which the quantities in eq. (I.) p. 96, refer ; 
but the same symbols, in the equation now demonstrated, are referable also to the ecliptic 
and its pole, the moon's orbit and its pole, the horizon and zenith, or to any imaginable 
great circle in the heavens at the pleasure of the calculator. 

Having demonstrated eq. (I.) to be thus general in its application, eq. (II.) Art. 96, be- 
comes by the manner of deduction from eq. I, equally universal ; and the conclusion at 
which we arrive immediately after eq. (II.) in Art. 97, may be stated without limitation 
as follows : 

" If, then, any two great circles cut one another at right angles at the point of the star 
to be occulted, p and q will be the cosines of the arcs joining the poles of these two cir- 
cles with the true place of the moon," &c. 



TABLES. 



137 



TABLE I. 

Changes of the circular functions in sign and algebraic expression through the 

four quadrants. 



o 

s 

CO 


© 


+■ 


o 


8 


7- 


8 


o 


4 1 


o" 

o 

■<3 do 

1 


3 g 

J ^ 

© i 

.S o 

CO cj 

1 1 


i g 

o <^ 

§ i 

co, <J 

to ^H 

3 •§ 


J 6 

© l 

I 8 
l 1 


< s 

I t- 

o 1 
oo 3 

11 

1 1 


o ■< 

to ^ 

CO o 
w CD 
O «2 

S 8 
■f-f 


CO | 

CD 

O CD 
CJ CO 

1 1 


6^ 

' 7 

o _li 

«o — • 

00 -j 

CD O 
> CJ 


o~" 

<1 o 

"& c 

S "§ 

o « 
o p» 

1 1 


o 

i 


7 


o 


8 


o 


8 


7 


+ * 


o' 

o 

&> § 
j 


1 ? 

00 1 

1-1 o 

i 3 

.5 o 

<o O 
1 1 


If 

1 © 
s .a 

cj to 

1 1 


1? 

is 


At 

+ + 


« 3" 

1 C* 

3 « 

cd 

O M 

a; O 

CO o 

1 1 


o ^-~ 
00 <^ 

if- 

CD "^ 

03 O 

O CD 
O to 

1 1 


3 

— to 
"S3 « 

01 o 

1 1 


o 

«< oo 

«i .g 

'co p 

£ o 

K» CJ 

1 1 


o 

I— 1 


o 


7 


o 


8 


1 


8 


-T 


7- 


o' 

^ J! 


J § 

O 1 

li 

•S o 

03 O 


o o> 
© 1 

§ .a 

V to 
1 1 


§ r 

J 8 

I 8 

1 1 


1 ° 

8 1 
I 1 


J* 

° ^ 

00 ^ 
0) 

O CO 

CD O 

1 1 


< 

if 

o ^. 

CD ^^ 
co CD 
O CD 

U to 

+ + 


^^ o 

<1 o 

J * 

o < 
oo -^ 

e g 
'53 « 

S cS 

> o 

1 1 

CO t» 


< 

if 

£ d 
cd '33 

8 1 

+■ + 


o 

8 


i- 


o 


8 


o 


8 


7- 


7- 


o 


o' 

^ s 

bO S 

s e 

« o 

« 1 
,1 


< 

M 

30 

4- 


< 

m 
o 


d 


< 

o 
o 


< 

(3 

CD 
CO 

+■ 


o 

CD 

O 
CD 

+ 


< 

a 

'to 
CD 

4; 


< 
.a 

CD 

o 

CJ 


o 

o 


o 


7- 


o 


8 


7- 


8 


o 


7- 




< 

© 

.a 

to 


<< 
§ 


< 


< 

o 


< 

O 

02 


< 
o 

CD 
CO 

o 


d 

oS 


1 

CD 

>■ 

o 



18 



138 



TABLES. 



TABLE II. 

For converting Intervals of Mean Solar Time into Equivalent Intervals of 
Sidereal Time. 



HOURS. 


MINUTES. 


SECONDS. 


h 


h 


m 


s 


m 


m 


s 


m 


m 


s 


m 


m 


8 


s 


s 


s 


8 


s 


8 


1 


1 





9.86 


1 


1 


0.16 


21 


21 


3.45 


41 


41 


6.74 


1 


1.00 


21 


21.06 


41 


41.11 


a 


2 





19.71 


2 


2 


0.33 


22 


22 


3.61 


42 


42 


6.90 


2 


2.01 


22 


22.06 


42 


42.11 


3 


3 





29.57 


3 


3 


0.49 


23 


23 


3.78 


43 


43 


7.06 


3 


3.01 


23 


23.06 


43 


43.12 


4 


4 





39.43 


4 


4 


0.66 


24 


24 


3.94 


44 


44 


7.23 


4 


4.01 


24 


24.07 


44 


44.12 


5 


5 





49.28 


5 


5 


0.82 


25 


25 


4.11 


45 


45 


7.39 


5 


5.01 


25 


25.07 


45 


45.12 


6 


6 





59.14 


G 


6 


0.99 


26 


26 


4.27 


46 


46 


7.56 


6 


6.02 


26 


26.07 


46 


46.13 


7 


7 


1 


9.00 


7 


7 


1.15 


27 


27 


4.44 


47 


47 


7.72 


7 


7.02 


27 


27.07 


47 


47.13 


8 


8 


1 


18.85 


8 


8 


1.31 


28 


28 


4.60 


48 


48 


7.89 


8 


8.02 


28 


23.08 


48 


48.13 


9 


9 


1 


28.71 


9 


9 


1.48 


29 


29 


4.76 


49 


49 


8.05 


9 


9.02 


29 


29.08 


49 


49.13 


10 


10 


1 


38.56 


10 


10 


1.64 


30 


30 


4.93 


50 


50 


8.21 


10 


10.03 


30 


30.08 


50 


50.14 


11 


11 


1 


48.42 


11 


11 


1.81 


31 


31 


5.09 


51 


51 


8.38 


11 


11.03 


31 


31.08 


51 


51.14 


1-2 


12 


1 


58.23 


12 


12 


1.97 


32 


32 


5.26 


52 


52 


8.54 


12 


12.03 


32 


32.09 


52 


52.14 


13 


13 


•2 


8.13 


13 


13 


2.14 


33 


33 


5.42 


53 


53 


8.71 


13 


13.04 


33 


33.09 


53 


53.15 


14 


14 


2 


17.99 


14 


14 


2.30 


34 


34 


5.59 


54 


54 


8.87 


14 


14.04 


34 


34.09 


54 


54.15 


15 


15 


2 


27.85 


15 


15 


2.46 


35 


35 


5.75 


55 


55 


9.04 


15 


15.04 


35 


35.10 


55 


55.15 


16 


16 


2 


37.70 


16 


16 


2.63 


36 


36 


5.91 


56 


56 


9.20 


16 


16.04 


36 


36.10 


56 


56.15 


17 


17 


2 


47.56 


17 


17 


2.79 ! 


37 


37 


6.03 


57 


57 


9.36 


17 


17.05 


37 


37.10 


57 


57.16 


18 


18 


2 


57.42 


[8 


IS 


2.96! 


33 


38 


6.24 


58 


58 


9.53 


18 


18.05 


38 


38.10 


58 


53.16 


19 


19 


3 


7.27 


19 


19 


3.12 39 


39 


6.41 


59 


59 


9.69 


! 9 


19.05 


39 


39.11 


59 


59.16 


20 
•21 


20 
21 


3 
3 


17.13 

26.99 


20 


20 


3.29 I 40 


40 


6.57 


60 


60 


9.86 


20 


20.05 


40 


40.11 


60 


60.16 












■22 


22 


3 


36.84 












23 


23 


3 


46.70 












2-1 


24 


3 


56.56 













TABLE III. 

For converting Intervals of Sidereal Time into Equivalent Intervals of Mean 

Solar Time. 



HOURS. 


MINUTES. 


SECONDS. 


h 


h m s 


m 


m s 


m 


m s 


m 


m s 


s 


8 


s 


8 


8 


8 


1 


59 50.17 


1 


59.84 


21 


20 56.56 


41 


40 53.28 


1 


1.00 


21 


20.94 


41 


40.89 


2 


1 59 40.34 


2 


1 59.67 


22 


21 56.40 


42 


41 53.12 


2 


1.99 


22 


21.94 


42 


41.89 


3 


2 59 30.51 


3 


2 59.51 


23 


22 56.23 


43 


42 52.96 


3 


2.99 


23 


22.94 


43 


42.83 


4 


3 59 20.63 


4 


3 59.34 


24 


23 56.07 


44 


43 52.79 


4 


3.99 


24 


23.93 


44 


43.88 


5 


4 59 10.85 


5 


4 59.18 


25 


24 55.90 


45 


44 52.63 


5 


4.99 


25 


24.93 


45 


44.88 


6 


5 59 1.02 


6 


5 59.02 


26 


25 55.74 


46 


45 52.46 


6 


5.98 


26 


25.93 


46 


45.87 


7 


6 53 51.19 


7 


6 58.85 


27 


26 55.58 


47 


46 52.30 


7 


6.98 


27 


26.93 


47 


46.87 


8 


7 53 41.36 


8 


7 58.69 


28 


27 55.41 


48 


47 52.14 


8 


7.98 


23 


27.92 


48 


47.87 


9 


8 58 31.53 


9 


8 58.53 i 29 


28 55.25 


49 


48 51.97 


9 


8.98 


29 


28.92 


49 


48.87 


10 


9 58 21.70 


10 


9 58.36 j 30 


29 55.09 


50 


49 51.81 


10 


9.97 


30 


29.92 


50 


49.86 


11 


10 58 11.87 


11 


10 58.20 31 


30 54.92 


51 


50 51.64 


11 


10.97 


31 


30.92 


51 


50.86 


12 


11 53 2.05 


12 


11 58.03 


32 


31 54.76 


52 


51 51.48 


12 


11.97 


32 


31.91 


52 


51.86 


13 


12 57 52.22 


13 


12 57.87 


33 


32 54.59 


53 


52 51.32 


13 


12.96 


33 


32.91 


53 


52.86 


14 


13 57 42.39 


14 


13 57.71 


34 


33 54.43 


54 


53 51.15 


14 


13.96 


34 


33.91 


54 


53.85 


15 


14 57 32.56 


15 


14 57.54 


35 


34 54.27 


55 


54 50.99 


15 


14.96 


35 


34.90 


55 


54.85 


16 


15 57 22.73 


16 


15 57.38 


36 


35 54.10 


56 


55 50.83 


16 


15.96 


36 


35.90 


56 


55.85 


17 


16 57 12.90 


17 


16 57.21 


37 


36 53.94 


57 


56 50.66 


17 


16.95 


37 


36.90 


57 


56.84 


18 


17 57 3.07 


18 


17 57.05 


38 


37 53.77 


58 


57 50.50 


18 


17.95 


33 


37.90 


58 


57.84 


19 


18 56 53.24 


19 


18 56.89 


39 


33 53.61 


59 


58 50.33 


19 


18.95 


39 


33.89 


59 


58.84 


26 


19 56 43.41 


20 


19 56.72 


40 


39 53.45 


60 


59 50.17 


20 


19.95 


40 


39.89 


60 


59.84 


2! 


20 56 33.58 






22 


21 56 23.75 






23 


22 56 13.92 






24 


23 56 4.09 







TABLES. 



139 



TABLE IV. 

Interpolation by Second Differences. 



■s.s 


to x 


Nat. 


CO- 


Log. coeffi- 
cient of d". 


SECONDS OF SECOND DIFFERENCES. 


a 
•- 




efficient 
'of d". 


10" 

1" 


20" 


30" 
3" 


40" 

4" 


50" 
5" 


GO" 
0" 


70" 
7" 


80" 
8" 


90" 

9" 


100" 
10" 




•te. 




^ 


a, S 


^ 




^ 


"~ 


q 


.1 

0.0 


.2 

0.1 


.3 
0.1 


.4 

0.2 


.5 

0.2 


.6 

0.3 


.7 
0.3 


.8 
0.4 


.9 
0.4 


1" 

0.5 


1 
m 
59.4 


.99 


.01 


m 
0.6' 


.0049 


49 
48 
47 
16 
45 
44 
13 
42 
41 
40 
39 
39 
37 
36 
35 
34 
33 
3-2 
31 
30 
29 
28 
27 
26 
25 
21 
23 


7.6946 


2966 

1717 

1204 

924 

745 

624 

532 

465 

409 

366 

328 

298 

272 

249 

229 

211 

196 

181 

169 

157 

147 

137 

128 

120 

112 

105 

98 

91 

86 

154 

134 

115 

96 

81 

65 

49 

35 

21 


.02 


1.2 


.0098 


7.9912 


0.1 


0.2 


0.3 


0,4 


0.5 


0.6 


0.7 


0.8 


0.9 


1.0 


58.8 


.98 


.03 


1.8 


.0145 


8.1629 


0.1 


0.3 


0.4 


0.6 


0.7 


0.9 


1.0 


1.2 


1.3 


1.5 


58.2 


.97 


.04 


2.4 


.0192 


8.2833 


0.2 


0.4 


0.6 


0*8 


1.0 


1.2 


1.3 


1.5 


1.7 


1.9 


57.6 


.96 


.05 


3.0 


.023? 


8.3757 


0.2 


0.5 


0.7 


0.9 


1.2 


l.l 


1.7 


1.9 


2.1 


2.4 


57.0 


.95 


.06 


3.6 


.0282 


8.4502 


0.3 


o.o 


0.8 


1.1 


1.1 


1.7 


2.0 


2.3 


2.5 


2.8 


56.4 


.94 


.07 


4.2 


.0325 


8.5126 


0.3 


0.7 


1.0 


1.3 


1.0 


2.0 


2.3 


2.6 


2.9 


3.3 


55.8 


.93 


.08 


4.8 


.0368 


8.5658 


0.4 


0.7 


1.1 


1.5 


1.8 


2.2 


2.6 


2.9 


3.3 


3.7 


55.2 


.92 


.09 


5.4 


.0409 


8.6123 


0.4 


0.8 


1.2 


1.6 


2.0 


2.5 


2.9 


3.3 


3.7 


4.1 


51.6 


.91 


.10 


6.0 


.0450 


8.6532 


0.5 


0.9 


1.4 


1,S 


2.2 


2.7 


3.1 


3.6 


4.1 


4.5 


54.0 


.90 


.11 


6.6 


.0489 


8.6398 


0.5 


1.0 


1.5 


2.0 


2.1 


2.0 


3.4 


3.9 


4.4 


4.9 


53.4 


.89 


.12 


7.2 


.0528 


8.7226 


0.5 


1.1 


1.6 


2.1 


2.6 


3.2 


3.7 


4.2 


4.8 


5.3 


52.8 


.88 


.13 


7.8 


.0565 


8.7524 


0.6 


1.1 


1.7 


2.3 


2.8 


3.4 


4.0 


4.5 


5.1 


5.7 


52.2 


.87 


.14 


8.4 


.0602 


8.7796 


0.6 


1.2 


1.8 


2.4 


3.0 


3.0 


4.2 


4.8 


5.4 


6.0 


51.6 


.86 


.15 


9.0 


.0637 


8.8045 


0.6 


1.3 


1.9 


2.5 


3.2 


3.8 


4.4 


5.1 


5.7 


6.4 


51.0 


.85 


.16 


9.6 


.0672 


8.8274 


0.7 


1.3 


2.0 


2.7 


3.1 


4.0 


4.6 


5.4 


6.0 


6.7 


50.4 


.84 


.17 


10.2 


.0705 


8.8485 


0.7 


1,1 


2.1 


2.8 


3.5 


1.2 


4.9 


5.6 


6.3 


7.1 


49.8 


.83 


.18 


10.8 


.0738 


8.8681 


0.7 


1.5 


2.2 


3.0 


3.7 


4.4 


5.2 


5.9 


6.6 


7.4 


49.2 


.82 


.19 


11.4 1 


.0769 


8.8862 


0.8 


1.5 


2.3 


3.1 


3.8 


i.O 


5.4 


6.2 


6.9 


7.7 


48.6 


.81 


.20 


12.0 


.0800 


8.9031 


0.8 


1.0 


2.4 


3.2 


4.0 


L8 


5.6 


6,4 


7.2 


8.0 


48.0 


.80 


.21 


12.6 


.0829 


8.9188 


0.8 


1.0 


2.5 


3.3 


4.1 


5.0 


5.8 


6.6 


7.5 


8.3 


47.4 


.79 


.22 


13.2 


.0858 


8.9335 


0.9 


1.7 


2.6 


3.1 


1.3 


5.1 


6.0 


6.9 


7.7 


8.6 


46.8 


.78 


.23 


13.8 


.0865 


8.9472 


0.9 


1.8 


2.7 


3.5 


4.4 


5.3 


6.2 


7.1 


8.0 


8.9 


46.2 


.77 


.24 


14.4 


.0912 


8.9600 


0.9 


l.s 


2.7 


3-..; 


1.0 


5.5 


6.4 


7.3 


8.2 


9.1 


45.6 


.76 


.25 


15.0 


.0937 


8.9720 


0.9 


i.g 


2.8 


3.7 




5.0 


6.6 


7.5 


8.4 


9.4 


45.0 


.75 


.26 


15.6 


.0962 


8.9832 


1.0 


1.9 


2.9 


3.8 


l.s 


5.8 


6.7 


7.7 


8.6 


9.6 


44.4 


.74 


.27 


16.2 


.0985 


8.9937 


1.0 


2.0 


3.0 


3.9 


1.9 


5.9 


6.9 


7.9 


8.9 


9.9 


43.8 


.73 


.28 


16.8 


.1008 


9.0035 


1.0 


2.0 


3.0 


1.0 


5.' 


6.0 


7.1 


8.1 


9.1 


10.1 


43.2 


.72 


.29 


17.4 


.1029 


21 
38 

31 
30 
26 
24 
18 
14 
10 
6 
2 


9.0126 


1.0 


2.1 


3.1 


i.i 


5.1 


0.2 


7.2 


8.3 


9.3 


10.3 


42.6 


.71 


.30 


18.0, 


.1050 


9.0212 


1.1 


2.1 


3.2 


1.2 


5.2 


0.3 


7.4 


8.4 


9.5 


10.5 


42.0 


.70 


.32 


19.2 


.1088 


9.0366 


1.1 


2.2 


3.3 


1.1 


5.; 


6.5 


7.6 


8.7 


9.8 


10.9 


40.8 


.68 


.34 


20.4 


.1122 


9.0500 


1.1 


2.2 


3.4 


4.5 


5.0 
5.8 


0., 




9.0 


10.1 


11.2 




M 


.36 


21.6 


.1152 


9.0615 


1.2 


2.3 


3.5 


4.6 


6.9 


8.0 


9.2 


10.1 


11.5 


38.4 


.64 


.38 


22.8 


.1178 


9.07 1 1 


1.2 


2.4 


3.5 


4.7 


5.9 


7.i 




9.4 


10.6 


11.8 


37.2 


.62 


.40 


24.0 


.1200 


9.0792 


1.2 


2.4 


3.6 


4.8 


0.0 




8.4 


9.0 


10.8 


12.0 


30.0 


.60 


.42 


25.2 


.1218 


9.0857 


1.2 


2.1 


3.7 


1.9 


o.: 


7.3 


8.5 




11.0 


12.-' 


34.8 


.58 


.44 


20. 1 


.1232 


9.0906 


1.2 


2.5 


3.7 


■1.9 


6.2 


7.! 


8.6 


9.'. 


11.1 


12.3 


33.0 


.56 


.46 


27.6 


.1242 


9.09 1 1 


1.2 


2.5 


3.7 


5.0 


6.2 


7.5 


8.7 


9.2 


11.2 


12.4 


32. 1 


.54 


.48 


28.8 


.1248 


9.0962 


1.2 


2.5 


3.7 


5.0 


0.2 


7.5 


8.7 


10.0 


11.2 


12.5 


31.2 


.52 


.50 


30.0 


.1250 


9.0969 


i 


1.3 


2.5 


3.8 


5.0 


0.3 


7.5 


8.8 


10.0 


11.3 


12.5 


30.0 


.50 



140 



TABLES. 



TABLE V. 



Coefficients of the errors a, b, and c, for the Transit Instrument, corresponding 
to every degree of Declination from the zenith of New Haven to the southern 
horizon. 



>. 



4-41' 

40 

39 

33 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

21 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

- 1 
2 

- 3 



a. 



0.007 
0.030 
0.052 
0.073 
0.094 
0.114 
0.134 
0.153 
0.172 
0.191 
0.209 
0.226 
0.244 
0.261 
0.277 
0.294 
0.310 
0.326 
0.341 
0.356 
0.371 
0.3S6 
0.401 
0.416 
0.430 
0.445 
0.459 
0.473 
0.487 
0.500 
0.514 
0.528 
0.541 
0.554 
0.568 
0.531 
0.594 
0.608 
0.621 
0.634 
0.647 
0.660 
0.673 
0.686 
0.699 



b. 



1.325 
1.305 
1.236 
1.267 
1.219 
1.231 
1.213 
1.196 
1.180 
1.164 
1.148 
1.132 
1.117 
1.102 
1.087 
1.073 
1.059 
1.045 
1.032 
1.018 
1.005 
0.991 
0.978 
0.966 
0.953 
0.941 
0.928 
0.916 
0.904 
0.892 
0.880 
0.868 
0.856 
0.844 
0.832 
0.821 
0.809 
0.797 
0.786 
0.774 
0.763 
0.751 
0.740 
0.728 
0.717 



1.325 
1.305 
1.287 
1.269 
1.252 
1.236 
1.221 
1.206 
1.192 
1.179 
1.167 
1.155 
1.143 
1.133 
1.122 
1.113 
1.103 
1.095 
1.086 
1.078 
1.071 
1.064 
1.058 
1.052 
1.046 
1.040 
1.035 
1.031 
1.026 
1.022 
1.019 
1.015 
1.012 
1.010 
1.008 
1.006 
1.004 
1.002 
1.001 
1.001 
1.000 
1.000 
1.000 
1.001 
1.001 



^ 



- 4° 
5 
6 
7 
8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 



a. 



0.712 
0.726 
0.739 
0.752 
0.765 
0.779 
0.792 
0.806 
0.820 
0.833 
0.847 
0.861 
0.876 
0.890 
0.904 
0.918 
0.933 
0.948 
0.963 
0.979 
0.994 
1.010 
1.026 
1.043 
1.059 
1.076 
1.094 
1.111 
1.129 
1.148 
1.167 
1.186 
1.206 
1.226 
1.247 
1.268 
1.290 
1.313 
1.336 
1.360 
1.385 
1.411 
1.438 
1.466 
1.494 



b. 



0.705 
0.694 
0.682 
0.670 
0.653 
0.647 
0.635 
0.623 
0.611 
0.599 
0.587 
0.574 
0.562 
0.549 
0.537 
0.524 
0.511 
0.498 
0.485 
0.471 
0.457 
0.443 
0.429 
0.415 
0.400 
0.335 
0.370 
0.355 
0.339 
0.323 
0.306 
0.239 
0.272 
0.254 
0.236 
0.217 
0.197 
0.177 
0.157 
0.136 
0.114 
0.091 
0.068 
0.043 
0.018 



C. 



1.002 
1.004 
1.006 
1.008 
1.010 
1.012 
1.015 
1.019 
1.022 
1.026 
1.031 
1.035 
1.040 
1.046 
1.052 
1.053 
1.064 
1.071 
1.078 
1.086 
1.095 
1.103 
1.113 
1.122 
1.133 
1.143 
1.155 
1.167 
1.179 
1.192 
1.206 
1.221 
1.236 
1.252 
1.269 
1.287 
1.305 
1.325 
1.346 
1.367 
1.390 
1.414 
1.440 
1.466 
1.494 



TABLES. 



141 



TABLE VI. 

Angles of the vertical with the earth's radius, and reduction of the equatorial 
horizontal parallax for every degree of latitude. 

Compression of the earth = ^}j . 





a 


£ ^ - 






e 


s- ^* 




•3 


VI t . 


COO 


^ 


•^ 




BOO 


^H 


3 

a 

H 






^ 

& 

q 


^3 

1 


43^ O- °9 


©S3 


^ 

$ 

Q 


1° 


0' 23".3 


0".00 


0".C0 


46° 


11' 8".8 


5".82 


0".10 


2 


46 .5 


.01 


.00 


47 


11 7 .7 


6 .01 


.10 


3 


1 9 .7 


.03 


.00 


48 


11 5 .7 


6 .21 


.11 


4 


1 32 .8 


.06 


.00 


49 


11 2 .9 


6 .41 


.11 


5 


1 55 .8 


.09 


.00 


50 


10 59 .3 


6 .59 


.11 


6 


2 18 .7 


.12 


.00 


51 


10 54 .9 


6 .79 


.12 


7. 


2 41 .4 


.17 


.00 


52 


10 49 .7 


6 .98 


.12 


8 


3 3 .8 


.22 


.00 


53 


10 43 .8 


7 .18 


.12 


9 


3 26 .1 


.23 


.00 


54 


10 37 .0 


7 .36 


.13 


10 


3 48 .1 


.34 


.01 


55 


10 29 .5 


7 .55 


.13 


11 


4 9 .9 


.41 


.01 


56 


10 21 .2 


7 .74 


.13 


12 


4 31 .4 


.48 


.01 


57 


10 12 .0 


7 .92 


.14 


13 


4 52 .5 


.57 


.01 


53 


10 2 .3 


8 .10 


.14 


14 


5 13 .2 


.65 


.01 


59 


9 51 .7 


8 .28 


.14 


15 


5 33 .7 


.75 


.01 


60 


9 40 .4 


8 .45 


.15 


16 


5 53 .6 


.85 


.01 


61 


9 28 .4 


8 .63 


.15 


17 


6 13 .2 


.96 


.02 


62 


9 15 .7 


8 .79 


.15 


18 


6 32 .3 


1 .06 


.02 


63 


9 2 .2 


8 .95 


.15 


19 


6 50 .9 


1 .18 


.02 


64 


8 48 .3 


9 11 


.16 


20 


7 9 .0 


1 .31 


.02 


65 


8 33 .7 


9 .26 


.16 


21 


7 26 .7 


1 .44 


.02 


66 


8 18 .3 


9 .41 


.16 


22 


7 43 .7 


1 .57 


.03 


67 


8 2 .4 


9 .55 


.16 


23 


8 .2 


1 .71 


.03 


68 


7 45 .9 


9 .70 


.17 


24 


8 16 .2 


1 .85 


.03 


69 


7 28 .8 


9 .83 


.17 


25 


8 31 .5 


2 .00 


.03 


70 


7 11 .1 


9 .97 


.17 


26 


8 46 .2 


2 .15 


.04 


71 


6 53 .0 


10 .08 


.17 


27 


9 .3 


2 .31 


.04 


72 


6 34 .3 


10 .21 


.18 


23 


9 13 .7 


2 .46 


.04 


73 


6 15 .2 


10 .32 


.18 


29 


9 26 .5 


2 .63 


.05 


71 


5 55 .6 


10 .43 


.18 


30 


9 33 .5 


2 .80 


.05 


75 


5 35 .6 


10 .54 


.18 


31 


9 40 .9 


2 .98 


.05 


76 


5 15 .0 


10 .64 


.18 


32 


10 .6 


3 .15 


.05 


77 


4 51 .2 


10 .72 


.19 


33 


10 10 .5 


3 .32 


.06 


78 


4 33 .0 


10 .80 


.19 


34 


10 19 .7 


3 .51 


.06 


79 


4 11 .4 


10 .88 


.19 


35 


10 28 .0 


3 .69 


.06 


80 


3 49 .6 


10 .94 


.19 


36 


10 35 .7 


3 .87 


.07 


81 


3 27 .4 


11 .01 


.19 


37 


10 4-> .6 


4 .06 


.07 


82 


3 5 .0 


11 .07 


.19 


33 


10 48 .7 


1 .-J.'. 


.07 


83 


2 42 .4 


11 .12 


.19 


39 


10 54 .1 


4 .45 


.08 


84 


2 19 .6 


11 .16 


.19 


40 


10 58 .6 


4 .64 


.08 


85 


1 56 .6 


11 .20 


.19 


41 


It 2 .3 


4 .83 


.08 


86 


1 33 .4 


11 .23 


.19 


42 


11 5 .2 


5 .03 


.09 


87 


1 10 .2 


11 .26 


19 


43 


11 7 .4 


5 .22 


.09 


88 


46 .8 


11 .27 


.19 


44 


11 8 .6 


5 .42 


.09 


89 


23 .4 


11 .29 


.19 


45 


11 9 .1 


5 .62 


.10 


90 


.0 


11 .29 


.19 



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